❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚
❖ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡
❇❛♥❛❝❤ ❡ ❛❧❣✉♠❛s ❆♣❧✐❝❛çõ❡s
†♣♦r
❈í❝❡r♦ ❉❡♠étr✐♦ ❱✐❡✐r❛ ❞❡ ❇❛rr♦s
s♦❜ ♦r✐❡♥t❛çã♦ ❞♦
Pr♦❢✳ ❉r✳ ▼❛♥❛ssés ❳❛✈✐❡r ❞❡ ❙♦✉③❛
❚r❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡ ❝✉rs♦ ❛♣r❡s❡♥✲ t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✲ ✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦✲ ♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡✲ q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
❆❣♦st♦✴✷✵✶✸ ❏♦ã♦ P❡ss♦❛ ✲ P❇
†❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡
❇❛♥❛❝❤ ❡ ❛❧❣✉♠❛s ❆♣❧✐❝❛çõ❡s
♣♦r
❈í❝❡r♦ ❉❡♠étr✐♦ ❱✐❡✐r❛ ❞❡ ❇❛rr♦s
❚r❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡ ❝✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✲ ✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ▼❛t❡♠át✐❝❛✳ ❆♣r♦✈❛❞❛ ♣♦r✿
Pr♦❢✳ ❉r✳ ▼❛♥❛ssés ❳❛✈✐❡r ❞❡ ❙♦✉③❛ ✲ ❯❋P❇ ✭❖r✐❡♥t❛❞♦r✮
Pr♦❢✳ ❉r✳ ❊✈❡r❛❧❞♦ ❙♦✉t♦ ❞❡ ▼❡❞❡✐r♦s ✲ ❯❋P❇
Pr♦❢✳ ❉r✳ ❏♦sé ❆♥❞❡rs♦♥ ❱❛❧❡♥ç❛ ❈❛r❞♦s♦ ✲ ❯❋❙
❉❡❞✐❝♦ ❛ ❉❡✉s ♣♦r t❡r ♠❡ ❝♦♥❝❡❞✐❞♦ ❞✐s❝❡r♥✐♠❡♥t♦ ❛ ✜♠ ❞❡ ♠❡ ❣✉✐❛r ♥❡ss❡ tr❛❜❛❧❤♦✳
❉❡❞✐❝♦ ❛ ♠❡✉ ■r♠ã♦ ❏♦sé ❉❛♥✐❧♦ ❱✐❡✐r❛ ❞❡ ❇❛rr♦s q✉❡✱ ❛♣❡s❛r ❞❡ ♥ã♦ ❡st❛r ♠❛✐s ❡♥tr❡ ♥ós✱ ❢♦✐ ❡ s❡♠♣r❡ s❡rá ❡①❡♠♣❧♦ ❞❡ ❢♦rç❛ ❡ s✉♣❡r❛çã♦✳
❉❡❞✐❝♦ ❛♦s ♠❡✉s ♣❛✐s ❏♦sé ❊❞♠✐❧s♦♥ ❋✐r♠✐♥♦ ❞❡ ❇❛rr♦s ❡ ▼❛r✐❛ ❉❡♠étr✐❛ ❱✐❡✐r❛ ❞❡ ❇❛rr♦s ❛ q✉❡♠ ❞❡✈♦ t♦❞♦ ♦ s✉❝❡ss♦ ❛❧❝❛♥ç❛❞♦ ❛té ❛q✉✐✳
❉❡❞✐❝♦ ❛ ♠✐♥❤❛ ❛♠❛❞❛ ❡s♣♦s❛ ❑íss✐❛ ❙✉♠❛②❛ ❋❡✐t♦s❛ ❈♦✉t✐♥❤♦ ❇❛rr♦s ♣♦r ♠❡ ❡♥t❡♥❞❡r ♥♦s ♠♦♠❡♥t♦s ❞❡ r❡♥ú♥❝✐❛ ❡♠ ♣r♦❧ ❞❛ ♣r♦❞✉çã♦ ❞❡ss❡ tr❛❜❛❧❤♦✳
❉❡❞✐❝♦ ❛♦s ♠❡✉s ✜❧❤♦s ❏♦ã♦ ●❛❜r✐❡❧ ❋❡✐t♦s❛ ❈♦✉t✐♥❤♦ ❇❛rr♦s✱ ❆♥❛ ❘✐t❛ ❋❡✐✲ t♦s❛ ❈♦✉t✐♥❤♦ ❇❛rr♦s ❡ ▼❛r✐❛ ❆❧✐❝❡ ❋❡✐t♦s❛ ❈♦✉t✐♥❤♦ ❇❛rr♦s ❛ q✉❡♠ ♠❡ ❞♦♦ ♣♦r ❝♦♠♣❧❡t♦✳
❉❡❞✐❝♦ ❛ t♦❞♦s ❛q✉❡❧❡s q✉❡ ❛❝r❡❞✐t❛r❛♠ q✉❡ ❛ r❡❛❧✐③❛çã♦ ❞❡ss❡ ♣r♦❥❡t♦✱ ❡♠ ❡s✲ ♣❡❝✐❛❧ ❛♦s ♠❡✉s ❝♦❧❡❣❛s ❞♦ P❘❖❋▼❆❚✴❯❋P❇ ♣❡❧♦s ♠♦♠❡♥t♦s ✈✐✈✐❞♦s ❛♦ ❧♦♥❣♦ ❞♦ ♥♦ss♦ ♠❡str❛❞♦ ❡ ♣❡❧♦ ❝r❡s❝✐♠❡♥t♦ ❝♦♥tí♥✉♦ ❞❡ ❛♣r❡♥❞✐③❛❣❡♠✳
❉❡❞✐❝♦ ❛♦s ❝♦❧❡❣❛s ♣r♦❢❡ss♦r❡s ▼❛r❝❡❧♦ ❉❛♥t❛s ❡ ●✉st❛✈♦ ❆ss❛❞ q✉❡ ♠❡ ❛❝♦♠♣❛✲ ♥❤❛r❛♠ ❞❡ ♣❡rt♦ ❡ ❥✉♥t♦s ❝♦♠✐❣♦ ❝♦♠♣❛rt✐❧❤❛r❛♠ ❞❛s ❡♠♦çõ❡s ❣❡r❛❞❛s ♥❛ r❡❛❧✐③❛çã♦ ❞❡ss❡ tr❛❜❛❧❤♦✳
❉❡❞✐❝♦ ❛♦s ♣r♦❢❡ss♦r❡s ❏♦ã♦ ▼❛r❝♦s ❇❡③❡rr❛ ❞♦ Ó✱ ❋❧á✈✐❛ ❏❡rô♥✐♠♦✱ ❇r✉♥♦ ❍❡♥✲ r✐q✉❡ ❡ ❝♦♠ ✉♠ t♦♠ ❡s♣❡❝✐❛❧ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ▼❛♥❛ssés ❳❛✈✐❡r ❞❡ ❙♦✉③❛ ❛ q✉❡♠ ❞❡✈♦ ♠✉✐t♦ ♣❡❧❛ ❛t❡♥çã♦✱ ♣❛❝✐ê♥❝✐❛ ❡ ♣r❡s❡♥ç❛ ❝♦♥st❛♥t❡ ❡♠ t♦❞♦ ♦ ♣r♦❝❡ss♦ ❞❡ ♣r♦❞✉çã♦ ❞❡ss❡ tr❛❜❛❧❤♦✳
◆❡ss❡ tr❛❜❛❧❤♦ ❢❛r❡♠♦s ✉♠❛ ❛❜♦r❞❛❣❡♠ s♦❜r❡ ❛ t❡♦r✐❛ ❞♦s ❡s♣❛ç♦s ♠étr✐❝♦s ❛ ✜♠ ❞❡ ❛♣r❡s❡♥t❛r♠♦s ♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤ ❡ ❡♠ s❡❣✉✐❞❛ ♦ ❛♣❧✐❝❛r❡♠♦s ❡♠ r❡s♦❧✉çõ❡s ❞❡ ❛❧❣✉♠❛s ❡q✉❛çõ❡s ♥ã♦ ❧✐♥❡❛r❡s ❝♦♠ ✉♠ ♠ét♦❞♦ ✐t❡r❛t✐✈♦ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞❛ s♦❧✉çã♦✳ ❋✐♥❛❧✐③❛r❡♠♦s ❛♣r❡s❡♥t❛♥❞♦ três ❛♣❧✐❝❛çõ❡s ❞♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤✳ ❆ ♣r✐♠❡✐r❛ s❡ tr❛t❛ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❊①✐stê♥❝✐❛ ❡ ❯♥✐❝✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s✳ ❆ s❡❣✉♥❞❛ t❡♠ ❝♦♠♦ t❡♠❛ ❛ ❛♣❧✐❝❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤ ♥❛ ár❡❛ ❞❡ ❝♦♠♣r❡ssã♦ ❞❡ ✐♠❛❣❡♥s ♥❛ ✐♥t❡r♥❡t✳ ❏á ❛ t❡r❝❡✐r❛ ❛♣❧✐❝❛çã♦ s❡rá ❛♣r❡s❡♥t❛❞♦ ❝♦♠♦ ❢✉♥❝✐♦♥❛ ♦ ❜✉s❝❛❞♦r ❞♦ ●♦♦❣❧❡ ❡ q✉❛❧ é ❛ ❝❛✉s❛ ❞♦ s❡✉ s✉❝❡ss♦✳
✶ ❖ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤ ✶ ✶✳✶ ▼♦t✐✈❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❊s♣❛ç♦s ▼étr✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✸ ❈♦♥✈❡r❣ê♥❝✐❛ ❡♠ ❊s♣❛ç♦s ▼étr✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✹ ❙❡q✉ê♥❝✐❛s ❞❡ ❈❛✉❝❤② ❡ ❊s♣❛ç♦s ▼étr✐❝♦s ❈♦♠♣❧❡t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✺ ❊①❡♠♣❧♦s ❞❡ ❊s♣❛ç♦s ▼étr✐❝♦s ❈♦♠♣❧❡t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✻ ❖ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✼ ❆❧❣✉♠❛s ❆♣❧✐❝❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✽ ❆❧❣✉♠❛s ❖❜s❡r✈❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷ ❖✉tr❛s ❆♣❧✐❝❛çõ❡s ❞♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤ ✸✵ ✷✳✶ ❚❡♦r❡♠❛ ❞❡ P✐❝❛r❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✷ ❈♦♠♣r❡ssã♦ ❞❡ ✐♠❛❣❡♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✸ ❈♦♠♦ ❢✉♥❝✐♦♥❛ ♦ ❜✉s❝❛❞♦r ❞♦ ●♦♦❣❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✹✺
◆❡ss❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛r❡♠♦s ♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤ ❝♦♠ ♦s ♣ré✲r❡q✉✐s✐t♦s ♣❛r❛ ❛ s✉❛ ❞❡♠♦♥str❛çã♦ ❡ ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s✱ ♣❛r❛ t❛♥t♦✱ ❞✐✈✐❞✐♠♦s ♥♦ss♦ tr❛❜❛❧❤♦ ❡♠ ❞♦✐s ❝❛♣ít✉❧♦s✳
◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ s❡rá ❛❜♦r❞❛❞❛ ✐♥✐❝✐❛❧♠❡♥t❡ ❛ t❡♦r✐❛ ❞♦s ❊s♣❛ç♦s ▼étr✐❝♦s ❝♦♠ ❛ ❛♣r❡s❡♥t❛çã♦ ❞❡ ❛❧❣✉♥s ❡①❡♠♣❧♦s ❝❧áss✐❝♦s ❞❡ ❊s♣❛ç♦s ▼étr✐❝♦s✱ ❝♦♠♦ s❡ ❞á ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❡♠ ❊s♣❛ç♦s ▼étr✐❝♦s ❝♦♠ ♦s ✐♠♣♦rt❛♥t❡s ❝♦♥❝❡✐t♦s ❞❡ s❡q✉ê♥❝✐❛s ❞❡ ❈❛✉❝❤② ❡ ❊s♣❛ç♦s ▼étr✐❝♦s ❈♦♠♣❧❡t♦s ❡ ✜♥❛❧✐③❛r❡♠♦s ♦ ❝❛♣ít✉❧♦ ❝♦♠ ❛ ❡①♣♦s✐çã♦ ❞♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤ s❡❣✉✐❞♦ ❞❡ ✉♠❛ ❝✉r✐♦s❛ ❛♣❧✐❝❛çã♦ ♣❛r❛ ♦❜t❡♥çã♦ ❞❡ r❛í③❡s ❞❡ ❢✉♥çõ❡s r❡❛✐s✿ ♦ ▼ét♦❞♦ ❞❡ ◆❡✇t♦♥ s♦❜ ❛ ❧✉③ ❞♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤✳
❋✐♥❛❧✐③❛r❡♠♦s ♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ❝♦♠ ✉♠ ❝♦♠❡♥tár✐♦ s♦❜r❡ ♦ t❡♦r❡♠❛ ❞♦ ♣♦♥t♦ ✜①♦ ❞❡ ❇r♦✉✇❡r ❡♠ ✉♠❛ ❞✐♠❡♥sã♦ ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ❡①✐❜✐r ♦✉tr❛ ❛❧t❡r♥❛t✐✈❛ ♥❛ ♦❜t❡♥çã♦ ❞❡ s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ♥ã♦ ❧✐♥❡❛r❡s✱ ❝♦♠♣❛r❛♥❞♦ ❛s ❝♦♥✈❡r❣ê♥❝✐❛s ❡♥tr❡ ♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤ ❡ ♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇r♦✉✇❡r✳
❋✐♥❛❧✐③❛r❡♠♦s ❡ss❡ tr❛❜❛❧❤♦ ❝♦♠ ♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♥❞♦ três ❛♣❧✐❝❛çõ❡s ❞♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤✳ ❆ ♣r✐♠❡✐r❛ s❡ tr❛t❛ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❊①✐stê♥❝✐❛ ❡ ❯♥✐❝✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s✳ ❆ s❡❣✉♥❞❛ t❡♠ ❝♦♠♦ t❡♠❛ ❛ ❛♣❧✐❝❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤ ♥❛ ár❡❛ ❞❡ ❝♦♠♣r❡ssã♦ ❞❡ ✐♠❛❣❡♥s ♥❛ ✐♥t❡r♥❡t✳ ❏á ❛ t❡r❝❡✐r❛ ❛♣❧✐❝❛çã♦ s❡rá ❛♣r❡s❡♥t❛❞♦ ❝♦♠♦ ❢✉♥❝✐♦♥❛ ♦ ❜✉s❝❛❞♦r ❞♦ ●♦♦❣❧❡ ❡ q✉❛❧ é ❛ ❝❛✉s❛ ❞♦ s❡✉ s✉❝❡ss♦✳
❖ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤
❯♠ t❡♦r❡♠❛ ❞❡ ♣♦♥t♦ ✜①♦ é ❡ss❡♥❝✐❛❧♠❡♥t❡ ✉♠ r❡s✉❧t❛❞♦ q✉❡ ❡st❛❜❡❧❡❝❡ ❝♦♥❞✐✲ çõ❡s ♣❛r❛ q✉❡ ❡①✐st❛ ✉♠ ❡❧❡♠❡♥t♦ x ❞♦ ❞♦♠í♥✐♦ ❞❡ ✉♠❛ ❛♣❧✐❝❛çã♦ f :M → M✱ t❛❧ q✉❡ f(x) = x✳
❙❡rá q✉❡ t♦❞❛ ❢✉♥çã♦ t❡♠ ♣♦♥t♦ ✜①♦❄ ◆❛ ♣rát✐❝❛✱ ❞❛❞❛ ✉♠❛ ❢✉♥çã♦f✱ ❞❡ ❝❡rt❛ ❢♦r♠❛ é ❞✐❢í❝✐❧ ❛ss❡❣✉r❛r✲s❡ ❞❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ♣♦♥t♦ ✜①♦ ♣❛r❛ f✳ ❊♠ ❛❧❣✉♠❛ s✐t✉❛çã♦✱ ♦ ✐♥t❡r❡ss❡ é ❣❛r❛♥t✐r ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ❡ s♦♠❡♥t❡ ✉♠ ♣♦♥t♦ ✜①♦ ♣❛r❛ ✉♠ ❢✉♥çã♦ ❞❛❞❛✳ ❖ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤ t❡♠ ❝♦♠♦ ♣❛rt✐❝✉❧❛r✐❞❛❞❡✱ ❛❧é♠ ❞❡ ❣❛r❛♥t✐r ❛ ❡①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❞♦ ♣♦♥t♦ ✜①♦ ❛✐♥❞❛ ❢♦r♥❡❝❡ ✉♠ ♣r♦❝❡ss♦ ✐t❡r❛t✐✈♦ q✉❡ ♣❡r♠✐t❡ ❡♥❝♦♥trá✲❧♦✳
❖ ♦❜❥❡t✐✈♦ ❝❡♥tr❛❧ ❞❡ss❡ ❝❛♣ít✉❧♦ é ♦ ❞❡ ❛♣r❡s❡♥t❛r✱ ❞❡ ❢♦r♠❛ s✉❝✐♥t❛✱ ❛s ❢❡r✲ r❛♠❡♥t❛s ❜ás✐❝❛s s♦❜r❡ ❡s♣❛ç♦s ♠étr✐❝♦s ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤✳ ❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ ❛♣❧✐❝❛çã♦ ✈❡r❡♠♦s ❛❧❣✉♥s r❡✲ s✉❧t❛❞♦s ❝♦♠♦ ❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ♥ã♦ ❧✐♥❡❛r❡s ❡ ♦ ❢❛♠♦s♦ ♠ét♦❞♦ ❞❡ ❞❡ ◆❡✇t♦♥ ♣❛r❛ ♦❜t❡♥çã♦ ❞❡ r❛í③❡s ❞❡ ✉♠❛ ❢✉♥çã♦ r❡❛❧✱ ♣♦r ❛♣r♦①✐♠❛çã♦ ♥✉♠ér✐❝❛✳
✶✳✶ ▼♦t✐✈❛çã♦
❙✉♣♦♥❤❛ q✉❡ ❏♦ã♦ ●❛❜r✐❡❧✱ ❛❧✉♥♦ ❞♦ ✶➸ ❛♥♦ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ♣❛r❛ ❡①♣❡r✐♠❡♥t❛r s✉❛ ♥♦✈❛ ❝❛❧❝✉❧❛❞♦r❛ ❝✐❡♥tí✜❝❛✱ ♣r❡s❡♥t❡ ❞❡ s❡✉ ♣❛✐✱ ❞❡❝✐❞❡ ♣r❡ss✐♦♥❛r ❛s t❡❝❧❛s ✏1✑ ❡
✏=✑✱ ♦❜t❡♥❞♦ ♦ ✈❛❧♦r1✳ ❉❡♣♦✐s ♣r❡ss✐♦♥♦✉ ❛ t❡❝❧❛ ✏cos✑ ♦❜t❡♥❞♦ ♦ ✈❛❧♦r0,5403023059
✭♣♦r ❛❝❛s♦ ❛ ❝❛❧❝✉❧❛❞♦r❛ ❞♦ ❏♦ã♦ ●❛❜r✐❡❧ ❡st❛✈❛ ❛ tr❛❜❛❧❤❛r ❡♠ r❛❞✐❛♥♦s✮✳ ❏♦ã♦ ●❛❜r✐❡❧ ❣♦st♦✉ ❞♦ r❡s✉❧t❛❞♦ ♦❜t✐❞♦ ❡✱ ♥✉♠ r❛r♦ í♠♣❡t♦ ❡①♣❡r✐♠❡♥t❛❧✐st❛✱ r❡s♦❧✈❡ ✈♦❧t❛r ❛ ♣r❡ss✐♦♥❛r s✉❝❡ss✐✈❛♠❡♥t❡ ❛ t❡❝❧❛ ✏cos✑ ♣♦r ❢♦r♠❛ ❛ ❝❛❧❝✉❧❛r ♦ ❝♦ss❡♥♦ ❞♦
♥ú♠❡r♦ ♣r❡✈✐❛♠❡♥t❡ ❝❛❧❝✉❧❛❞♦❀ ❞❡s✐❣♥❛♥❞♦ ♣♦r n ♦ ♥ú♠❡r♦ ❞❡ ✈❡③❡s q✉❡ ❛ t❡❝❧❛ ✏cos✑ ❢♦✐ ♣r❡ss✐♦♥❛❞❛ ❡ ♣♦r xn ✈❛❧♦r ♦❜t✐❞♦✱ t❡♠✲s❡✿
n xn n xn n xn n xn
1 0,5403023059 16 0,7395672022 31 0,739083847 46 0,7390851366 2 0,8575532158 17 0,7387603199 32 0,7390859996 47 0,7391301765 3 0,6542897905 18 0,7393038924 33 0,7390845496 48 0,7390851309 4 0,7934803587 19 0,7389377567 34 0,7390855264 49 0,7390851348 5 0,7013687736 20 0,7391843998 35 0,7390848684 50 0,7390851322 6 0,7639596829 21 0,7390182624 36 0,7390853116 51 0,7390851339 7 0,7221022425 22 0,7391301765 37 0,739085013 52 0,7390851327 8 0,7504177618 23 0,7390547907 38 0,7390852142 53 0,7390851335 9 0,7314040424 24 0,7391055719 39 0,7390850787 54 0,739085133 10 0,7442373549 25 0,7390713653 40 0,7390851699 55 0,7390851334 11 0,7356047404 26 0,7390944074 41 0,7390851085 56 0,7390851331 12 0,7414250866 27 0,739078886 42 0,7390851499 57 0,7390851333 13 0,7375068905 28 0,7390893414 43 0,739085122 58 0,7390851332 14 0,7401473356 29 0,7390822985 44 0,7390851408 59 0,7390851332 15 0,7383692041 30 0,7390870427 45 0,7390851281 60 0,7390851332
❈❤❡❣❛❞♦ ❛ ❡st❡ ♣♦♥t♦✱ ❞❡♣♦✐s ❞❡ ♣r❡ss✐♦♥❛r ✏✻✵✑ ✈❡③❡s ❛ t❡❝❧❛ ✏cos✑✱ ♦ ❏♦ã♦ ●❛❜r✐❡❧
❛ss✉st♦✉✲s❡ ✭❛rr❡♣❡♥❞❡♥❞♦✲s❡ ❛té ❞❡ s❡✉ í♠♣❡t♦✮✳ ❚❡r✲s❡✲✐❛ ❛ ❝❛❧❝✉❧❛❞♦r❛ ❛✈❛r✐❛❞❛❄ P♦r q✉❡ é q✉❡✱ ❞❡♣♦✐s ❞❡ ♣r❡ss✐♦♥❛r 58✈❡③❡s ❛ t❡❝❧❛ ✏cos✑✱ ♣♦r ♠❛✐s q✉❡ ❝♦♥t✐♥✉❛ss❡
❛ r❡♣❡t✐r ❛ ♦♣❡r❛çã♦✱ ♦❜t✐♥❤❛ s❡♠♣r❡ ♦ ♠❡s♠♦ ✈❛❧♦r❄ ❆✢✐t♦ ❡ ♥ã♦ ❡♥❝♦♥tr❛♥❞♦ ♦ ♥ú♠❡r♦ ❞❡ t❡❧❡❢♦♥❡ ❞❛ ✏❘❡♣❛r❛❞♦r❛ ❞❡ ❈❛❧❝✉❧❛❞♦r❛s✑✱ r❡s♦❧✈❡ ❢❛③❡r ✉♠❛ ❡①♣❡r✐ê♥❝✐❛ ❛♥á❧♦❣❛✱ ♠❛s ♣❛rt✐♥❞♦ ❞♦ ✈❛❧♦r 335 ❡♠ ✈❡③ ❞♦ ✈❛❧♦r 1✳ ❖❜t❡✈❡ ♥❛ 60 ✈❡③ x60 =
0,7390851332✳ ❆♦ ✜♠ ❞❡ ♣r❡♠✐r 57 ✈❡③❡s ❛ t❡❝❧❛ ✏cos✑ ♦❜t✐♥❤❛ ♦ ♠❡s♠♦ ✈❛❧♦r ❞❡
❤á ♣♦✉❝♦✳ ❉❡s❛❧❡♥t❛❞♦✱ ❏♦ã♦ ●❛❜r✐❡❧ r❡s♦❧✈❡✉ ❢❛③❡r ✉♠❛ ❡①♣❡r✐ê♥❝✐❛ ❛♥á❧♦❣❛ ❝♦♠ ❛ t❡❝❧❛ ✏tan−1✑ ✭❝♦rr❡s♣♦♥❞❡♥t❡ à ❢✉♥çã♦ ❛r❝♦ t❛♥❣❡♥t❡✮✱ ♣❛rt✐♥❞♦ ❞♦ ✈❛❧♦r 1❀ ❡✐s ♦s
r❡s✉❧t❛❞♦s✿
n xn n xn n xn n xn
1 0,7853981934 16 0,2957885899 31 0,2163331839 46 0,1786587773 2 0,66577375 17 0,2875886511 32 0,2130500906 47 0,1767935042 3 0,5873841757 18 0,2800317036 33 0,2099116578 48 0,1749853452 4 0,5310915102 19 0,2730381016 34 0,2069075819 49 0,1732314471 5 0,4882103378 20 0,2665413363 35 0,204028557 50 0,1715291523 6 0,4541714733 21 0,2604853784 36 0,2012661543 51 0,1698759817 7 0,4263175061 22 0,2548226492 37 0,198612719 52 0,1682696198 8 0,4029860575 23 0,2495124513 38 0,1960612806 53 0,166707901 9 0,3830779111 24 0,2445197411 39 0,193605477 54 0,165188797 10 0,3658337989 25 0,2398141588 40 0,1912394875 55 0,1637104065 11 0,3507104329 26 0,2353692536 41 0,1889579751 56 0,1622709445 12 0,3373075815 27 0,2311618604 42 0,1867560362 57 0,1608687339 13 0,325323103 28 0,2271715935 43 0,1846291561 58 0,1595021966 14 0,314524107 29 0,2233804326 44 0,1825731707 59 0,1581698468 15 0,3047278305 30 0,2197723811 45 0,180584238 60 0,1568702837
❖ ❏♦ã♦ ●❛❜r✐❡❧ ♥ã♦ ❡♥t❡♥❞❡✉ ♦ q✉❡ s❡ ♣❛ss❛✈❛✳ ❖❜t❡✈❡ ✉♠❛ s✉❝❡ssã♦ ❞❡ ♥ú✲ ♠❡r♦s ♣♦s✐t✐✈♦s q✉❡ ❛♣❛r❡♥t❡♠❡♥t❡ é ❞❡❝r❡s❝❡♥t❡❀ ♠❛s sê✲❧♦✲á ❞❡ ❢❛t♦❄ ❙❡rá q✉❡ t❡♥❞❡rá ♣❛r❛ ③❡r♦❄ ❙❡rá q✉❡ ❝♦♥✈❡r❣❡ ♣❛r❛ ♦✉tr♦ ❧✐♠✐t❡❄ P♦r q✉❡ é q✉❡ ❛ ♠áq✉✐♥❛ s❡ ❝♦♠♣♦rt❛ ❞❡ ♠❛♥❡✐r❛ ❞✐❢❡r❡♥t❡ ♥❡st❡ ú❧t✐♠♦ ❝❛s♦✱ ♥ã♦ ❡st❛❜✐❧✐③❛♥❞♦ ♦ r❡s✉❧t❛❞♦ ❝♦♠♦ ♥♦s ❝❛s♦s ❛♥t❡r✐♦r❡s❄ ❖ ❏♦ã♦ ●❛❜r✐❡❧ r❡s♦❧✈❡✉ ♣❡♥s❛r ✉♠ ♣♦✉❝♦ ❡✱ ❛♦ ❝❤❡❣❛r ❛ ❝♦♥❝❧✉sã♦ ♥❡♥❤✉♠❛✱ ❞❡❝✐❞✐✉ ♣❡r❣✉♥t❛r ❛♦ s❡✉ ♣❛✐ s♦❜r❡ ♦ ♦❝♦rr✐❞♦✱ q✉❡ ♣r♦♥t❛♠❡♥t❡ ❧❤❡ r❡s♣♦♥❞❡ q✉❡ t❛❧ ❛ss✉♥t♦ ♥ã♦ ❢❛③✐❛ ♣❛rt❡ ❞♦ ❝♦♥t❡ú❞♦ ♣r♦❣r❛♠át✐❝♦ ❞♦ ✶➸ ❛♥♦ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ♠❛s q✉❡ ❡❧❡ s❡ ❞❡♣❛r❛r✐❛✱ ❛♦ ❧♦♥❣♦ ❞❡ s✉❛ ✈✐❞❛ ❡st✉❞❛♥t✐❧ ❝♦♠ ❛❧❣✉♠❛s ❡q✉❛çõ❡s ❝❤❛♠❛❞❛s ❞❡ ✏❡q✉❛çõ❡s ♥ã♦ ❧✐♥❡❛r❡s✑ ❡ q✉❡✱ ❞❡ ✐♥í❝✐♦ t❡r✐❛♠ ♠ét♦❞♦s ❛❧t❡r✲ ♥❛t✐✈♦s ❝♦♠♦ ❝♦♠♣❛r❛çã♦ ❣rá✜❝❛✱ ♠ét♦❞♦s ✐t❡r❛t✐✈♦s ❝♦♠♦ ♦ ❞❛ ✏❜✐ss❡çã♦✑ ❡✱ ♣❛r❛ ❛❧✉♥♦s ♠❛✐s ❛✈❛♥ç❛❞♦s✱ ♣♦❞❡r✐❛♠ s❡r ❛♣r❡s❡♥t❛❞♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❞❡ ❚❡♦r❡♠❛s ❞❡ P♦♥t♦ ❋✐①♦✳ ✭❍✐stór✐❛ ❛❞❛♣t❛❞❛ ❞❛ ❘❡✈✐st❛ ●❛③❡t❛ ❞❡ ▼❛t❡♠át✐❝❛✱ ❥❛♥❡✐r♦ ✷✵✵✷ ✲ ♥➸ ✶✹✷✱ P♦rt✉❣❛❧✮
◆♦s ú❧t✐♠♦s ✺✵ ❛♥♦s ❛ t❡♦r✐❛ ❞♦s ♣♦♥t♦s ✜①♦s r❡✈❡❧♦✉✲s❡ ❝♦♠♦ ✉♠❛ ❢❡rr❛♠❡♥t❛ ♣♦❞❡r♦s❛ ❡ ✐♠♣♦rt❛♥t❡ ♥♦ ❡st✉❞♦ ❞♦s ❢❡♥ô♠❡♥♦s ♥ã♦ ❧✐♥❡❛r❡s✱ ♠❛♣❡❛♥❞♦ ❡ ✐♥❞✐❝❛♥❞♦ ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s✳ ▼❛t❡♠át✐❝♦s ❝♦♠♦ ❈❛✉❝❤②✱ ▲✐♦✉✈✐❧❧❡✱ ▲✐♣s❝❤✐t③✱ P❡❛♥♦✱ ❋r❡❞❤♦❧♠✱ P✐❝❛r❞✱ ❇❛♥❛❝❤✱ ❇r♦✉✇❡r✱ ❍❡❧♣❡r♥✱ ▼❛♥♥✱ ■s❤✐❦❛✇❛✱ ❡t❝✱ ❞❡r❛♠ ❞✐❢❡r❡♥t❡s t✐♣♦s ❞❡ ♠ét♦❞♦s ✐t❡r❛t✐✈♦s ♣❛r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ❞❡ ♣♦♥t♦ ✜①♦✳
✶✳✷ ❊s♣❛ç♦s ▼étr✐❝♦s
◗✉❛♥❞♦ s❡ ❢❛③ ✉♠ ❝✉rs♦ ❞❡ ❝á❧❝✉❧♦✱ s❡❥❛ ❡♠ ✉♠❛ ♦✉ ❡♠ ✈ár✐❛s ✈❛r✐á✈❡✐s✱ s❡♠♣r❡ ♥♦s ❞❡♣❛r❛♠♦s ❝♦♠ ❛ ♥♦çã♦ ❞❡ ❞❡r✐✈❛çã♦ ❡ ✐♥t❡❣r❛çã♦✱ ❝❡♥tr❛❞❛s ♥❛ ♥♦çã♦ ✐♥t✉✐t✐✈❛ ❞❡ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♣♦♥t♦s✱ s❡❥❛ ❡♠ ❢♦r♠❛ ❞❡ ✐♥t❡r✈❛❧♦s ♥♦ ❝❛s♦ ❞❛ r❡t❛ r❡❛❧✱ s❡❥❛ ♥❛ ❢♦r♠❛ ❞❡ ✈✐③✐♥❤❛♥ç❛✱ ♥♦ ❝❛s♦ ❞❡ ♠❛✐s ❞❡ ✉♠❛ ✈❛r✐á✈❡❧✳ ❆ss✐♠✱ ♣♦r ❡①❡♠♣❧♦✱ q✉❛♥❞♦ ✈❛♠♦s tr❛t❛r ❞❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s(xn)♣❛r❛
✉♠ ♣♦♥t♦ a✱ ❞❡✈❡♠♦s ♥♦s ❝❡rt✐✜❝❛r q✉❡ |xn−a| ✜❝❛ ❝❛❞❛ ✈❡③ ♠❛✐s ♣ró①✐♠♦ ❞❡ ③❡r♦
à ♠❡❞✐❞❛ q✉❡ n ❝r❡s❝❡✱ ♦✉ s❡❥❛✱ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ❡❧❡♠❡♥t♦s ❞❛ s❡q✉ê♥❝✐❛ ❡ s❡✉ ❧✐♠✐t❡ ✈❛✐ ✜❝❛♥❞♦ ❝❛❞❛ ✈❡③ ♠❡♥♦r✳
❆♦ ❧♦♥❣♦ ❞❛ s✉❛ ❡✈♦❧✉çã♦✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❞❡♣♦✐s ❞♦ sé❝✉❧♦ ❳■❳✱ ❛ ▼❛t❡♠át✐❝❛ ❣❛♥❤♦✉ ✉♠❛ r♦✉♣❛❣❡♠ ❜❡♠ ♠❛✐s ❛❜str❛t❛ ❣❡♥❡r❛❧✐③❛♥❞♦ ❛ ♥♦çã♦ ❞❡ ❞✐stâ♥❝✐❛ ❞❡ t❛❧ ❢♦r♠❛ ❛ ❛♣❧✐❝á✲❧❛ ❛ ❝♦♥❥✉♥t♦s ♥ã♦ tã♦ ❝♦♠✉♥s ❝♦♠♦ R✱R2 ♦✉ R3 ✳ ❊s♣❛ç♦s ❝♦♠ ❞✐♠❡♥sã♦ ♠❛✐♦r ❡ ❛té ❞❡ ❞✐♠❡♥sã♦ ✐♥✜♥✐t❛ ❣❛♥❤❛r❛♠ ❛t❡♥çã♦ ♥♦ ❡st✉❞♦ ❞❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♣♦♥t♦s ❞♦ ❡s♣❛ç♦✳ ❆ss✐♠✱ t❛✐s ❡st✉❞♦s ❝♦♥❞✉③✐r❛♠ às ♥♦çõ❡s ❞❡ ▼étr✐❝❛ ❡ ❞❡ ❊s♣❛ç♦s ▼étr✐❝♦s ✐♥✐❝✐❛❧♠❡♥t❡ ✐♥tr♦❞✉③✐❞❛s ♣♦r ▼❛✉r✐❝❡ ❘❡♥é ❋ré❝❤❡t ✭✶✽✼✽ ✲ ✶✾✼✸✮✱ ♠❛t❡♠át✐❝♦ ❢r❛♥❝ês q✉❡ ❛❧é♠ ❞❡ss❡ ❝♦♥❝❡✐t♦ t❛♠❜é♠ ❢♦✐ r❡s♣♦♥sá✈❡❧ ♣♦r ❡st❛❜❡❧❡❝❡r ♦s ❢✉♥❞❛♠❡♥t♦s ❞❛ t♦♣♦❧♦❣✐❛✱ ❝♦♥✈❡r❣ê♥❝✐❛ ✉♥✐❢♦r♠❡ ❛❧é♠ ❞❡ t❡r s✐❞♦ ♦ ♣r✐♠❡✐r♦ ❛ ✉s❛r ❛ ❡①♣r❡ssã♦ ✏❊s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✑✳
❆ss✐♠✱ q✉❛✐s s❡r✐❛♠ ♦s ✏✐♥❣r❡❞✐❡♥t❡s✑ ❛ s❡r❡♠ ✉s❛❞♦s ♣❛r❛ s❡ tr❛❞✉③✐r✱ ❞❡ ❢♦r♠❛ ♣r❡❝✐s❛✱ ❛ ♥♦çã♦ ❞❡ ❞✐stâ♥❝✐❛❄ P❛r❛ ✐ss♦✱ ❢❛ç❛♠♦s ❛
❉❡✜♥✐çã♦ ✶✳✶ ❙❡❥❛ M ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦✳ ❯♠❛ ❢✉♥çã♦ d : M ×M → R é ❝❤❛♠❛❞❛ ❞❡ ♠étr✐❝❛ ❡♠ M s❡ ❣♦③❛r ❞❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿
✶✳ d(x, y) = 0 ⇔ x=y ♣❛r❛ t♦❞♦ x, y ∈ M❀ ✷✳ d(x, y)>0✱ ♣❛r❛ t♦❞♦ x, y ∈ M ❡ x6=y❀
✸✳ d(x, y) = d(y, x)✱ ♣❛r❛ t♦❞♦ x, y ∈ M ✭s✐♠❡tr✐❛✮❀
✹✳ d(x, y)≤d(x, z) +d(z, y)✱ ♣❛r❛ t♦❞♦ x, y, z ∈ M ✭❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✮✳ ❆♦ ♣❛r (M, d) ❝❤❛♠❛♠♦s ❞❡ ❊s♣❛ç♦ ▼étr✐❝♦✳
❱❡❥❛♠♦s ❛ s❡❣✉✐r ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❡s♣❛ç♦s ♠étr✐❝♦s✳
❊①❡♠♣❧♦✿ ❙❡❥❛♠ M ✉♠ ❝♦♥❥✉♥t♦ q✉❛❧q✉❡r ♥ã♦ ✈❛③✐♦ ❡ ❝♦♥s✐❞❡r❡♠♦s ❛ ❢✉♥çã♦ d:M ×M →R ❞❡✜♥✐❞❛ ♣♦r
d(x, y) =
0, s❡ x=y
➱ ✐♠❡❞✐❛t❛ ❛ ✈❡r✐✜❝❛çã♦ ❞❡ q✉❡d é ✉♠❛ ♠étr✐❝❛✱ ❡ss❛ é ❝❤❛♠❛❞❛ ♠étr✐❝❛ ③❡r♦✲ ✉♠✳
❊①❡♠♣❧♦✿ ❈♦♥s✐❞❡r❡ M =R ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ❡ ❛ ❢✉♥çã♦ d:M ×M →R ❞❡✜♥✐❞❛ ♣♦r
d(x, y) =|x−y|.
❯s❛♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ♠ó❞✉❧♦ ❞❡ ✉♠ ♥ú♠❡r♦ r❡❛❧✱ s❡❣✉❡ q✉❡d é ✉♠❛ ♠étr✐❝❛✳ ❉❡ ❢❛t♦✱ s❡❥❛♠ x, y e z ∈R q✉❛✐sq✉❡r✳ ❙❡ x=y✱ é ✐♠❡❞✐❛t♦ q✉❡ |x−y|= 0✱ ♦ q✉❡
✐♠♣❧✐❝❛ ❡♠ d(x, y) = 0✳ ❆❧é♠ ❞✐ss♦✱ s❡ x 6= y✱ t❡♠♦s x−y 6= 0 ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠
|x−y|>0✱ ♦✉ s❡❥❛✱ q✉❡d(x, y)>0✳ ❚❛♠❜é♠✱ t❡♠♦s q✉❡
d(x, y) = |x−y|=| −(x−y)|=|y−x|=d(y, x). P♦r ✜♠✱ t❡♠♦s q✉❡
|x−z|=|(x−y) + (y−z)| ≤ |x−y|+|y−z|.
▲♦❣♦✱ d(x, z) ≤ d(x, y) +d(y, z)✳ P♦rt❛♥t♦✱ d é ✉♠ ♠étr✐❝❛ ❛ q✉❛❧ é ❝❤❛♠❛❞❛ ❞❡ ♠étr✐❝❛ ✉s✉❛❧ ❞❡ R✳
❊①❡♠♣❧♦✿ ❆❣♦r❛✱ ❝♦♥s✐❞❡r❛♥❞♦
M =RN ={(x1, x2, . . . , xN);xi ∈R ❡ i= 1,2. . . . , N}
❤á três ♠étr✐❝❛s ✐♠♣♦rt❛♥t❡s ❡♠ RN✳ P❛r❛ x= (x1, x2, . . . , xN) ❡
y= (y1, y2, . . . , yN)✱ ❝♦♥s✐❞❡r❡♠♦s✿
❼ de :RN ×RN →R ❞❡✜♥✐❞❛ ♣♦r
de(x, y) =
p
(x1−y1)2+ (x2−y2)2+· · ·+ (xN −yN)2.
Pr♦✈❛r❡♠♦s q✉❡ de é ✉♠❛ ♠étr✐❝❛✳ ❈♦♠ ❡❢❡✐t♦✱ s❡❥❛♠x, y, z ∈RN✳ ❆ss✐♠✱
de(x, x) =
p
(x1−x1)2+ (x2−x2)2+· · ·+ (xN −xN)2 =
√
02 + 02+· · ·+ 02 = 0.
❙❡ x 6= y✱ ❡♥tã♦ xi 6= yi ♣❛r❛ ❛❧❣✉♠ i ∈ {1,2,3, . . . , N}✳ ❆ss✐♠✱ ❝♦♠♦ xi −yi 6= 0
t❡♠♦s (xi−yi)2 >0✳ P♦rt❛♥t♦✱
de(x, y) =
p
(x1−y1)2+ (x2−y2)2+· · ·+ (xN −yN)2 ≥
p
(xi−yi)2 >0.
P❛r❛ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ s✐♠❡tr✐❛✱ s❛❜❡♠♦s q✉❡ (xi −yi)2 = (yi − xi)2✱ ♣❛r❛ t♦❞♦
i∈ {1,2,3, . . . , N}✳ ❉❛í✱ t❡♠♦s✿
(x1−y1)2+ (x2−y2)2+· · ·+ (xN−yN)2 = (y1−x1)2+ (y2−x2)2+· · ·+ (yN−xN)2.
❊①tr❛✐♥❞♦ ❛ r❛✐③ ❡♠ ❛♠❜♦s ♦s ❧❛❞♦s ❞❛ ✐❣✉❛❧❞❛❞❡✱ t❡♠♦s✿
p
(x1−y1)2+ (x2−y2)2+· · ·+ (xN −yN)2 =
p
(y1−x1)2+ (y2−x2)2+· · ·+ (yN −xN)2.
❉❛í✱ ❝♦♥❝❧✉í♠♦s q✉❡ de(x, y) = de(y, x)✳ ❋✐♥❛❧♠❡♥t❡✱ ♣❛r❛ ♠♦str❛r ❛ ❞❡s✐❣✉❛❧❞❛❞❡
tr✐❛♥❣✉❧❛r✱ s❡rá ♥❡❝❡ssár✐♦ ♦ ✉s♦ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ ❡♠RN✱ ❛ q✉❛❧
❞✐③ q✉❡ ♣❛r❛ t♦❞♦ a, b∈RN✱ ♦♥❞❡ a= (a1, . . . , aN)❡ b= (b1, . . . , bN)✱
|a1b1+a2b2 +· · ·+aNbN| ≤
q
a2
1+· · ·+a2N
q
b2
1+· · ·+b2N
. ✭✶✳✶✮ Pr♦✈❛ ❞❡ ✭✶✳✶✮✿ ❈♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦f :R→R❞❡✜♥✐❞❛ ♣♦rf(t) =||a−tb||2 ❝♦♠
a, b∈RN✱ ♦♥❞❡de(a, b) =ka−bk✱ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ha, bi=a1b1+a2b2+· · ·+aNbN
❡ ♥♦r♠❛||a||2 =ha, ai✳ ❱❡❥❛♠♦s q✉❡f(t)≥0✱∀t∈R✱ ♦✉ s❡❥❛✱ ||a−tb||2 ≥0✱ ♦ q✉❡
✐♠♣❧✐❝❛ q✉❡
||a−tb||2 =ha−tb, a−tbi=ha, ai −2tha, bi+t2hb, bi, ♦✉ s❡❥❛✱
||a||2−2tha, bi+t2||b||2 ≥0.
❈♦♠♦ ✐ss♦ só ♦❝♦rr❡ q✉❛♥❞♦ (−2ha, bi)2−4||a||2||b||2 ≤0✱ ♦✉ s❡❥❛✱
|ha, bi ≤ ||a|| · ||b||. ❆❣♦r❛✱ ♣r♦✈❛r❡♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✿
[de(x, z)]2 = N
X
i=1
(xi−zi)2
=
N
X
i=1
[(xi−yi) + (yi−zi)]2
=
N
X
i=1
(xi−yi)2+ 2(xi−yi)(yi−zi) + (yi−zi)2
=
N
X
i=1
(xi−yi)2+ 2 N
X
i=1
(xi−yi)(yi−zi) + N
X
i=1
(yi−zi)2
= [de(x, y)]2+ 2 N
X
i=1
(xi−yi)(yi−zi) + [de(y, z)]2
✉s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③✱
≤ [de(x, y)]2+ 2
v u u t
N
X
i=1
(xi−yi)2
v u u t
N
X
i=1
(yi−zi)2+ [de(y, z)]2.
= [de(x, y)]2+ 2de(x, y)·de(y, z) + [de(y, z)]2
= [de(x, y) +de(y, z)]2.
❆ss✐♠✱ de(x, z)≤de(x, y) +de(y, z)✳
❉❡ ♠❛♥❡✐r❛ ♠❛✐s s✐♠♣❧❡s✱ ♣♦❞❡✲s❡ ♠♦str❛r q✉❡ ❛s ❢✉♥çõ❡sds❡d∞❞❡s❝r✐t❛s ❛❜❛✐①♦
sã♦ ♠étr✐❝❛s ❡♠ RN✳
❼ ds :RN ×RN →R ❞❡✜♥✐❞❛ ♣♦r
ds(x, y) = |x1−y1|+|x2−y2|+· · ·+|xN −yN|.
❼ d∞ :RN ×RN →R❞❡✜♥✐❞❛ ♣♦r
d∞(x, y) = max{|x1−y1|,|x2−y2|, . . . ,|xN −yN|}.
❱❡❥❛♠♦s ❛❣♦r❛ ✉♠ ❡①❡♠♣❧♦ ❞❡ ❡s♣❛ç♦ ♠étr✐❝♦ ♠❛✐s ❛❜str❛t♦✳
❊①❡♠♣❧♦✿ ❙❡❥❛ M = C[a, b] = {f : [a, b]→R; ❢ é ❝♦♥tí♥✉❛ ❡♠[a, b]} ❝♦♠ ❛s
♠étr✐❝❛s ❼ d′ :M
×M ❞❡✜♥✐❞❛ ♣♦r
d′(f, g) = max
t∈[a,b]|f(t)−g(t)|.
▼♦str❛r❡♠♦s q✉❡ d′ é ✉♠❛ ♠étr✐❝❛✳ ❉❡ ✐♥í❝✐♦✱ ♣♦❞❡♠♦s ✈❡r q✉❡ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❲❡✐❡rstr❛ss ✭♣❛r❛ ❞❡t❛❧❤❡s✱ ✈❡r ❬✼❪✮✱ d′ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✳ ❙❡❥❛♠ f ❡ g ❢✉♥çõ❡s q✉❛✐sq✉❡r ❞❡ C[a, b]✳ ❆ss✐♠✱ t❡♠♦s q✉❡
d(f, f) = max
t∈[a,b]|f(t)−f(t)|= maxt∈[a,b]|0|= 0.
❙❡ f 6=g ❡♥tã♦ ❡①✐st❡ ✉♠ t0 ∈[a, b] t❛❧ q✉❡ f(t0)6=g(t0)✳
❆ss✐♠✱
0<|f(t0)−g(t0)| ≤ max
t∈[a,b]|f(t)−g(t)|=d(f, g).
❙❛❜❡♠♦s t❛♠❜é♠ q✉❡✱ ♣❛r❛ t♦❞♦ t ∈[a, b]✱ ✈❛❧❡
d(f, g) = max
t∈[a,b]|f(t)−g(t)|= maxt∈[a,b]|g(t)−f(t)|=d(g, f).
◆♦t❡ q✉❡
|f(t)−h(t)| = |f(t)−g(t) +g(t)−h(t)|
≤ |f(t)−g(t)|+|f(t)−h(t)|
≤ max
t∈[a,b]|f(t)−g(t)|+ maxt∈[a,b]|g(t)−h(t)|,
♣♦✐s
|f(t)−g(t)| ≤ max
t∈[a,b]|f(t)−g(t)|
|g(t)−h(t)| ≤ max
t∈[a,b]|g(t)−h(t)|.
❈♦♠ ✐ss♦✱ ❝♦♥❝❧✉í♠♦s ❢❛❝✐❧♠❡♥t❡ q✉❡✿
max
t∈[a,b]|f(t)−h(t)| ≤tmax∈[a,b]|f(t)−g(t)|+ maxt∈[a,b]|g(t)−h(t)|,
♦✉ s❡❥❛✱ d(f, h)≤d(f, g) +d(g, h)✳
❆♥❛❧♦❣❛♠❡♥t❡✱ ♣♦❞❡✲s❡ ♠♦str❛r q✉❡ ❼ d′′ :M ×M ❞❡✜♥✐❞❛ ♣♦r
d′′(f, g) =
Z b
a |
f(t)−g(t)|dt t❛♠❜é♠ é ✉♠❛ ♠étr✐❝❛✳
✶✳✸ ❈♦♥✈❡r❣ê♥❝✐❛ ❡♠ ❊s♣❛ç♦s ▼étr✐❝♦s
❉❡✜♥✐çã♦ ✶✳✷ ❯♠❛ s❡q✉ê♥❝✐❛ (xn) ❡♠ ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ (M, d) ❝❤❛♠❛✲s❡ ❧✐♠✐✲
t❛❞❛ q✉❛♥❞♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ s❡✉s t❡r♠♦s é ❧✐♠✐t❛❞♦✱ ✐st♦ é✱ q✉❛♥❞♦ ❡①✐st❡ c > 0 t❛❧
q✉❡ d(xm, xn)≤c ✱ ♣❛r❛ q✉❛✐sq✉❡r m, n∈N✳
P♦r ❡①❡♠♣❧♦ ❝♦♥s✐❞❡r❡ M = R✱ ❝♦♠ ❛ ♠étr✐❝❛ ✉s✉❛❧ ❡ ❛ s❡q✉ê♥❝✐❛ (xn)✱ t❛❧ q✉❡
xn = (−1)n✳ ❱❡♠♦s q✉❡ ♣❛r❛ q✉❛❧q✉❡r ♥❛t✉r❛❧ ♥✱ t❡♠♦s|(−1)n|= 1✳ ❉❛í✱ ❡s❝♦❧❤✐❞♦s
q✉❛✐sq✉❡r m ❡n ♥❛t✉r❛✐s✱ t❡♠♦s
d(xm, xn) =|(−1)m−(−1)n| ≤ |(−1)m|+|(−1)n| ≤1 + 1 = 2.
❉❡✜♥✐çã♦ ✶✳✸ ❯♠❛ s❡q✉ê♥❝✐❛(xn)❡♠ ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦(M, d)é ❞✐t❛ ❝♦♥✈❡r❣❡♥t❡
❡♠ M s❡ ❡①✐st✐r x∈M t❛❧ q✉❡ limn→∞d(xn, x) = 0✳ ■st♦ é✱
♣❛r❛ t♦❞♦ ε >0, ❡①✐st❡n0; ♣❛r❛ t♦❞♦ n > n0 t❡♠✲s❡ d(xn, x)< ε.
❆q✉✐✱ x é ❝❤❛♠❛❞♦ ❞❡ ❧✐♠✐t❡ (xn)✳ ◗✉❛♥❞♦ ❢♦r ♥❡❝❡ssár✐♦✱ ✉s❛r❡♠♦s ❛ ♥♦t❛çã♦
xn →x ♣❛r❛ ✐♥❞✐❝❛r ❛ ❝♦♥✈❡r❣ê♥❝✐❛✳
▲❡♠❛ ✶✳✶ ❙❡❥❛ (M, d) ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✳ ❊♥tã♦✱
✭❛✮ ❯♠❛ s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡ ❡♠ M é ❧✐♠✐t❛❞❛ ❡ s❡✉ ❧✐♠✐t❡ é ú♥✐❝♦✳ ✭❜✮ ❙❡ xn →x ❡ yn→y ❡♠ M✱ ❡♥tã♦ d(xn, yn)→d(x, y).
Pr♦✈❛✿
✭❛✮ ❙✉♣♦♥❤❛ q✉❡ xn → x ❡♠ M✳ ❊♥tã♦✱ ✜①❛♥❞♦ ε = 1✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r n0 t❛❧
q✉❡
d(xn, x)<1 ♣❛r❛ t♦❞♦ n > n0.
➱ ❡✈✐❞❡♥t❡ t❛♠❜é♠ q✉❡ ♣❛r❛ t♦❞♦ m≤n0 t❡♠♦s q✉❡
d(xm, x)≤max{d(x1, x), . . . , d(xn0, x)}.
❊♠ ❝♦♥s❡q✉ê♥❝✐❛ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r✿ d(xn, xm)≤d(xn, x) +d(x, xm),
❧♦❣♦✱
d(xn, xm)≤1 + max{d(x1, x), . . . , d(xn0, x)}.
❖ q✉❡ ♠♦str❛ q✉❡ (xn) é ❧✐♠✐t❛❞❛✳ ❆ss✐♠✱ ❜❛st❛ t♦♠❛r
c= 1 + max{d(x1, x), . . . , d(xn0, x)}.
❈♦♥s✐❞❡r❡♠♦s ❛❣♦r❛ q✉❡ xn → a ❡ q✉❡ xn → b ❡♠ M✳ ❉❛í✱ ♣❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡
tr✐❛♥❣✉❧❛r✱ t❡♠♦s✿
d(a, b)≤d(a, xn) +d(xn, b).
❚♦♠❛♥❞♦ ♦ ❧✐♠✐t❡ q✉❛♥❞♦ n→ ∞✱ t❡r❡♠♦sd(a, b)≤0✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡d(a, b) = 0
❡ ✐ss♦ ♦❝♦rr❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ a=b✳ ▲♦❣♦✱ ♦ ❧✐♠✐t❡ é ú♥✐❝♦✳
✭❜✮ P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✱ t❡♠♦s
d(xn, yn)≤d(xn, x) +d(x, y) +d(y, yn).
❆ss✐♠✱
d(xn, yn)−d(x, y)≤d(xn, x) +d(y, yn). ✭✶✳✷✮
P♦r ♦✉tr♦ ❧❛❞♦✱
d(x, y)≤d(x, xn) +d(xn, yn) +d(yn, y).
❖ q✉❡ ✐♠♣❧✐❝❛ ❡♠✱
−[d(x, xn) +d(yn, y)]≤d(xn, yn)−d(x, y). ✭✶✳✸✮
❉❡ ✭✶✳✷✮ ❡ ✭✶✳✸✮✱ t❡♠♦s q✉❡
|d(xn, yn)−d(x, y)| ≤d(xn, x) +d(y, yn).
❚♦♠❛♥❞♦ ♦ ❧✐♠✐t❡ q✉❛♥❞♦ n → ∞ ❡ ✉s❛♥❞♦ q✉❡
d(xn, x)→0 ❡ d(yn, y)→0,
♦❜t❡♠♦s q✉❡ d(xn, yn)→d(x, y)✳
✶✳✹ ❙❡q✉ê♥❝✐❛s ❞❡ ❈❛✉❝❤② ❡ ❊s♣❛ç♦s ▼étr✐❝♦s ❈♦♠✲
♣❧❡t♦s
❉❡✜♥✐çã♦ ✶✳✹ ❯♠❛ s❡q✉ê♥❝✐❛(xn)é ❞❡ ❈❛✉❝❤② ♥✉♠ ❊s♣❛ç♦ ▼étr✐❝♦(M, d)q✉❛♥❞♦
q✉❛❧q✉❡r q✉❡ s❡❥❛ ε > 0✱ é ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r n0 ∈ N t❛❧ q✉❡✱ ❡s❝♦❧❤✐❞♦s q✉❛✐sq✉❡r
❞♦✐s í♥❞✐❝❡s ❛ ♣❛rt✐r ❞❡ n0✱ ❞✐❣❛♠♦s n ❡ m✱ t❡r❡♠♦s s❡♠♣r❡ d(xn, xm)< ε✳
❯♠❛ ❢♦r♠❛ ❡q✉✐✈❛❧❡♥t❡ ❞❡ ♠♦str❛r q✉❡ ✉♠❛ s❡q✉ê♥❝✐❛✱ ♥❛s ♠❡s♠❛s ❝♦♥❞✐çõ❡s ❛❝✐♠❛✱ é ❞❡ ❈❛✉❝❤② é ❡s❝r❡✈❡r♠♦s ♦ í♥❞✐❝❡ m+n = p✱ p ∈ N ❡ ♠♦str❛r♠♦s q✉❡
lim
n→∞d(xn, xn+p) = 0✳
■ss♦ s✐❣♥✐✜❝❛ q✉❡ q✉❛♥❞♦ ♦s t❡r♠♦s ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ s❡ ❛♣r♦①✐♠❛♠ ❞❡ ✉♠ ♣♦♥t♦ ✜①❛❞♦✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❛♣r♦①✐♠❛♠✲s❡ ✉♥s ❞♦s ♦✉tr♦s✳
❉❡✜♥✐çã♦ ✶✳✺ ❯♠ ❡s♣❛ç♦ ♠étr✐❝♦ (M, d) é ❞✐t♦ s❡r ❝♦♠♣❧❡t♦ s❡ t♦❞❛ s❡q✉ê♥❝✐❛ ❞❡
❈❛✉❝❤②✱ t♦♠❛❞❛ ❡♠ M✱ ❝♦♥✈❡r❣✐r ❡♠ M✳
❊①❡♠♣❧♦✿ ❖ ❡s♣❛ç♦ Q ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ♥ã♦ é ❝♦♠♣❧❡t♦ q✉❛♥❞♦ ❞♦t❛❞♦ ❞❛ ♠étr✐❝❛ ✉s✉❛❧ ❡♠ R✳ P❛r❛ ✐st♦✱ ❝♦♥s✐❞❡r❡ ❛ s❡q✉ê♥❝✐❛ ❡♠ q✉❡ x1 = 1✱ x2 = 1 +
1 1!✱
x3 = 1+
1 1!+
1
2!✱x4 = 1+ 1 1!+
1 2!+
1
3!✱❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✳ Pr✐♠❡✐r♦✱ ✈❡❥❛♠♦s q✉❡ (xn) é ❞❡ ❈❛✉❝❤②✳ ❉❡ ❢❛t♦✱ d(xn, xn+p) =
1 (n+p)!
✱ ♦✉ s❡❥❛
lim
n→∞d(xn, xn+p) = 0✳ ❚❛♠❜é♠ s❛❜❡♠♦s✱ ♣❡❧♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❙ér✐❡ ❞❡ ❚❛②❧♦r✱ q✉❡xn →e✳ ◆♦ ❡♥t❛♥t♦✱
❝♦♠♦ e /∈Q ✭✈❡r ❡♠ ❬✸❪✮✱ t❡♠♦s q✉❡ Q ♥ã♦ é ❝♦♠♣❧❡t♦✳
Pr♦♣♦s✐çã♦ ✶ ❚♦❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ❡♠ (M, d) é ❧✐♠✐t❛❞❛✳
Pr♦✈❛✿ ❚♦♠❡♠♦s ✉♠❛ s❡q✉ê♥❝✐❛(xn)✱ ❞❡ ❈❛✉❝❤②✱ ❡♠(M, d)✳ ❆ss✐♠✱ ✜①❛♥❞♦ε= 1✱
❞❡✈❡ ❡①✐st✐r ✉♠ n0 ∈N✱ t❛❧ q✉❡✿
d(xm, xn)<1, ♣❛r❛ t♦❞♦ m, n > n0.
P❛r❛ m, n < n0✱ s❡❥❛ c= max{d(xm, xn);m, n≤n0}✱ ❡♥tã♦
d(xm, xn)≤c+ 1, ∀ m, n∈N.
Pr♦♣♦s✐çã♦ ✷ ❙❡❥❛ (xn) ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ❡♠ (M, d) ❡ (xnk) ✉♠❛ s✉❜✲
s❡q✉ê♥❝✐❛ ❞❡ (xn)✳ ❙❡ (xnk) é ❝♦♥✈❡r❣❡♥t❡✱ ❡♥tã♦ (xn) t❛♠❜é♠ é ❝♦♥✈❡r❣❡♥t❡✳
Pr♦✈❛✿ ❈♦♥s✐❞❡r❡ (xnk) ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❞❡ (xn) t❛❧ q✉❡ xnk → a✱ ❝♦♠ a ∈ M✳
❚♦♠❡ ε >0✳ ❈♦♠♦ (xn) é ❞❡ ❈❛✉❝❤②✱ ❞❡✈❡ ❡①✐st✐rn1 ∈N t❛❧ q✉❡
d(xm, xn)<
ε
2, ♣❛r❛ t♦❞♦ m, n > n1.
❈♦♠♦ xnk →a✱ ❞❡✈❡ ❡①✐st✐r ✉♠ n2 ∈N✱ t❛❧ q✉❡
d(xnk, a)<
ε
2, ♣❛r❛ t♦❞♦ nk > n2
❚♦♠❡♠♦s n0 = max{n1, n2} ✱ ❡s❝r❡✈❡♥❞♦ m = n0 + 1 ❡ ✉s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡
tr✐❛♥❣✉❧❛r✱ t❡♠♦s✿
d(xn, a)≤d(xn, xn0+1) +d(xn0+1, a)<
ε
2 +
ε
2 =ε, ♣❛r❛ t♦❞♦ n > n0.
❆ss✐♠✱ ❝♦♥❝❧✉í♠♦s q✉❡ xn→a✳
Pr♦♣♦s✐çã♦ ✸ ❆ r❡t❛ r❡❛❧✱ (R,| · |) ✱ é ✉♠ ❊s♣❛ç♦ ▼étr✐❝♦ ❈♦♠♣❧❡t♦✳
Pr♦✈❛✿ ❆❝❛❜❛♠♦s ❞❡ ❞❡♠♦♥str❛r q✉❡ t♦❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② é ❧✐♠✐t❛❞❛✳ ❆s✲ s✐♠✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❇♦❧③❛♥♦✲❲❡✐❡rstr❛ss✱ q✉❡ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ ❬✼❪✱ t♦❞❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s✱ ♣♦ss✉✐ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡✳ ❉❡ss❛ ❢♦r♠❛✱ ❝♦♠❜✐♥❛♥❞♦ ❛s ❞✉❛s ♣r♦♣♦s✐çõ❡s✱ ❝♦♥❝❧✉í♠♦s q✉❡ R é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦✳
❖❜s❡r✈❛çã♦✿ ❆♥❛❧♦❣❛♠❡♥t❡✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ [a, b] ❝♦♠ ❛ ♠étr✐❝❛ ✉s✉❛❧✱
Pr♦♣♦s✐çã♦ ✹ ❚♦❞❛ s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡ ❡♠ (M, d) é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤②✳
Pr♦✈❛✿ ❙❡ xn →a✱ ❡♥tã♦✱ ♣❛r❛ t♦❞♦ ε >0✱ ❡①✐st❡ n0 ∈N t❛❧ q✉❡
d(xn, a)<
ε
2, ♣❛r❛ t♦❞♦ n > n0.
❉❛í✱ ♣❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✱ ♣❛r❛ m, n > n0✱ t❡♠♦s✿
d(xn, xm)≤d(xn, a) +d(a, xm)<
ε
2+
ε
2 =ε.
▲♦❣♦✱ (xn) é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤②✳
❉❡✜♥✐çã♦ ✶✳✻ ❙❡❥❛♠(M, dM)❡(N, dN)❡s♣❛ç♦s ♠étr✐❝♦s✳ ❯♠❛ ❛♣❧✐❝❛çã♦f :M →
N é ❝♦♥tí♥✉❛ ♥♦ ♣♦♥t♦ a ∈M q✉❛♥❞♦✱ ♣❛r❛ t♦❞♦ ε >0 ❞❛❞♦✱ é ♣♦ssí✈❡❧ ♦❜t❡r δ >0
t❛❧ q✉❡
dM(x, a)< δ ⇒dN(f(x), f(a))< ε.
❉✐③✲s❡ q✉❡ f :M →N é ❝♦♥tí♥✉❛ q✉❛♥❞♦ ❡❧❛ é ❝♦♥tí♥✉❛ ❡♠ t♦❞♦s ♦s ♣♦♥t♦s ❞❡ M✳ Pr♦♣♦s✐çã♦ ✺ ❯♠❛ ❢✉♥çã♦ f : X →Y ❞❡✜♥✐❞❛ ❡♥tr❡ ♦s ❡s♣❛ç♦s ♠étr✐❝♦s (X, d1)
❡♠ (Y, d2) é ❝♦♥tí♥✉❛ ❡♠ ✉♠ ♣♦♥t♦a∈X s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♣❛r❛ q✉❛❧q✉❡r s❡q✉ê♥❝✐❛
(xn) ❡♠ X✱ t❡♠✲s❡✿
xn d1
−→a ⇒f(xn) d2
−→f(a).
Pr♦✈❛✿ ❆ss✉♠✐♥❞♦ q✉❡ f é ❝♦♥tí♥✉❛✱ t❡♠♦s q✉❡ ♣❛r❛ t♦❞♦ ε >0✱ ❡①✐st❡ δ > 0✱ t❛❧
q✉❡✱ ♣❛r❛ t♦❞♦ x∈X
d1(x, a)< δ ⇒d2(f(x), f(a))< ε.
❙❡❥❛ xn d1
−→a✳ ❊♥tã♦ ❡①✐st❡ n0 t❛❧ q✉❡ ♣❛r❛ t♦❞♦ n > n0 ✱ t❡♠♦s d1(xn, a)< δ✱ ♣❛r❛
t♦❞♦ ε >0✳ ❈♦♠♦ f é ❝♦♥tí♥✉❛✱ t❡♠♦s q✉❡
d1(xn, a)< δ ⇒d2(f(xn), f(a)< ε,
❡✱ ❡st❛ ú❧t✐♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❡q✉✐✈❛❧❡ ❞✐③❡r q✉❡ f(xn) d2
−→f(a) ✳
❆ss✉♠✐♥❞♦ ❛❣♦r❛ q✉❡ xn d1
−→ a ⇒ f(xn) d2
−→ f(a), ♣r♦✈❡♠♦s q✉❡ f é ❝♦♥tí♥✉❛ ❡♠ a ✳ ❙✉♣♦♥❤❛✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡ ✐ss♦ s❡❥❛ ❢❛❧s♦✳ ❊♥tã♦✱ ♣❛r❛ t♦❞♦ δ > 0✱ ❡①✐st❡
ε0 >0✱ ❡ ✉♠ x6=a s❛t✐s❢❛③❡♥❞♦
d1(x, a)< δ, ♠❛s d2(f(x), f(a))≥ε0.
❈♦♠♦ δ > 0 é q✉❛❧q✉❡r✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ♣❛r❛ ❝❛❞❛ n ∈ N✱ ❢❛ç❛♠♦s δ = 1
n✳ ❉❛í✱ t❡rí❛♠♦s✿
d1(xn, a)<
1
n ♠❛s d2(f(xn), f(a))≥ε0. ❆ss✐♠✱ ❝♦♥str✉í♠♦s ✉♠❛ s❡q✉ê♥❝✐❛ ❡♠ q✉❡ xn
d1
−→a✱ ♠❛s f(xn) ♥ã♦ ❝♦♥✈❡r❣❡ ♣❛r❛
f(a)✱ ❝♦♥tr❛❞✐③❡♥❞♦ ♥♦ss❛ ❤✐♣ót❡s❡✳
✶✳✺ ❊①❡♠♣❧♦s ❞❡ ❊s♣❛ç♦s ▼étr✐❝♦s ❈♦♠♣❧❡t♦s
❊①❡♠♣❧♦✿ ❖ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦ RN é ❝♦♠♣❧❡t♦ ❝♦♠ ❛s ♠étr✐❝❛sd
e, ds ❡d∞✳ ▼♦s✲
tr❛r❡♠♦s ❝♦♠ ❛ ♠étr✐❝❛ de✳
❈♦♠ ❡❢❡✐t♦✱
de(x, y) =
p
(x1−y1)2+ (x2−y2)2+· · ·+ (xN −yN)2.
❊♠ q✉❡ x = (x1, x2, . . . , xN) ❡ y = (y1, y2, . . . , yN) ✳ ❈♦♥s✐❞❡r❡♠♦s ✉♠❛ s❡q✉ê♥❝✐❛
❞❡ ❈❛✉❝❤② q✉❛❧q✉❡r (xm)❡♠ RN✱ t❡♠✲s❡xm = (x( m) 1 , x
(m) 2 , . . . , x
(m)
N )✳
❙❡♥❞♦ (xm) ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤②✱ t❡♠♦s ♣❛r❛ t♦❞♦ε >0, ❡①✐st❡ n0 ∈Nt❛❧ q✉❡
d(xm, xp) = N
X
k=1
(x(km)−x
(p)
k )2
! 1 2
< ε ❝♦♠ m, p > n0. ✭✶✳✹✮
❊❧❡✈❛♥❞♦ ✭✶✳✹✮ ❛♦ q✉❛❞r❛❞♦✱ ♦❜t❡♠♦s✿
N
X
k=1
(x(km)−xk(p))2< ε2, ♣♦rt❛♥t♦✱
|x(km)−xk(p)|< ε, ❝♦♠ m, p > n0.
❊ ❛ ✐♠♣❧✐❝❛çã♦ ♦❝♦rr❡✱ ♣♦✐s t❡♠♦s ✉♠❛ s♦♠❛ ❞❡ t❡r♠♦s ♣♦s✐t✐✈♦s ♠❡♥♦r q✉❡ ε2✳
❆ss✐♠✱ ❝❛❞❛ t❡r♠♦ é ♠❡♥♦r q✉❡ ε2✳ ❉❡st❛ ❢♦r♠❛✱ ♣❛r❛ ❝❛❞❛ k ✜①♦✱ t❡♠♦s q✉❡ ❛
s❡q✉ê♥❝✐❛(x(1)k , x(2)k , . . . ,)é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ❞❡ ♥ú♠❡r♦s r❡❛✐s✳ ❊❧❛ ❝♦♥✈❡r❣❡
♣❡❧❛ ♣r♦♣♦s✐çã♦ ✸✱ ❞✐❣❛♠♦sx(km)→xk✱ q✉❛♥❞♦m→ ∞✳ ❯s❛♥❞♦ ❡ss❡ ❧✐♠✐t❡N ✈❡③❡s✱
❞❡✜♥✐♠♦s x= (x1, x2, . . . , xn)∈RN✳ ❉❛ ❡①♣r❡ssã♦ ✭✶✳✹✮✱ t❡♠♦s✿
lim
m→∞d(xm, xp) = mlim→∞
" N X
k=1
x(km)−x(kp)2
#12
=
" N X
k=1
lim
m→∞x
(m)
k −mlim→∞x
(p)
k
2 #
1 2
=
" N X
k=1
xk−x( p)
k
2 #12
= d(x, xp)≤ε, ❝♦♠ m > n0.
■ss♦ ♠♦str❛ q✉❡ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② t♦♠❛❞❛ é ❝♦♥✈❡r❣❡♥t❡ ❡♠RN✳
❉❡✜♥✐çã♦ ✶✳✼ ❯♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❢✉♥çõ❡s fn : M → R ❝♦♥✈❡r❣❡ ✉♥✐❢♦r♠❡♠❡♥t❡
♣❛r❛ ❛ ❢✉♥çã♦ f :M →R q✉❛♥❞♦✱ ♣❛r❛ t♦❞♦ ε >0 ❞❛❞♦✱ ❡①✐t❡ n0 ∈N✱ ❞❡♣❡♥❞❡♥❞♦
❛♣❡♥❛s ❞❡ ε✱ t❛❧ q✉❡ n > n0 ⇒d(fn(x), f(x))< ε s❡❥❛ q✉❛❧ ❢♦r x∈M✳
❊①❡♠♣❧♦✿ ❖ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s C[a, b] é ❝♦♠♣❧❡t♦ ❝♦♠ ❛ ♠étr✐❝❛ d(f, g) = maxt∈[a,b]|f(t)−g(t)|.
❈♦♠ ❡❢❡✐t♦✱ s❡❥❛ (fn) ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ❡♠ C[a, b]✳ ❊♥tã♦✱ ❞❛❞♦ ε > 0✱
❡①✐st❡ n0 ∈N t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦m, n > n0✱ t❡♠♦s✿
d(fm, fn) = max
t∈[a,b]|fm(t)−fn(t)|< ε. ✭✶✳✺✮
❆ss✐♠✱ ♣❛r❛ t♦❞♦ t∈[a, b],
|fm(t)−fn(t)|< ε, ❝♦♠ m, n > n0.
❉❛í✱ t❡♠♦s q✉❡ (f1(t), f2(t), . . .) é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ❞❡ ♥ú♠❡r♦s r❡❛✐s✳ ❏á
q✉❡ (R,| · |) é ❝♦♠♣❧❡t♦✱ ❛ s❡q✉ê♥❝✐❛(fn(t)) ❝♦♥✈❡r❣❡✱ ❞✐❣❛♠♦s fn(t)→ f(t)✳ ❉❡ss❡
♠♦❞♦✱ ♣❡❧❛ ✉♥✐❝✐❞❛❞❡ ❞♦ ❧✐♠✐t❡✱ ♣♦❞❡♠♦s ❛ss♦❝✐❛r ♣❛r❛ ❝❛❞❛ t ∈ [a, b] ✉♠ ú♥✐❝♦
♥ú♠❡r♦ r❡❛❧ f(t)✳ ■ss♦ ❞❡✜♥❡ ✉♠❛ ❢✉♥çã♦ f ❡♠ [a, b]✳ ▼♦str❡♠♦s q✉❡f ♣❡rt❡♥❝❡ ❛ C[a, b]❡ q✉❡fn→f✱ ✉♥✐❢♦r♠❡♠❡♥t❡✳ ❉❛ ❡①♣r❡ssã♦ ✭✶✳✺✮✱ q✉❛♥❞♦n → ∞✱ ♦❜t❡♠♦s✿
ε ≥ lim
n→∞d(fm, fn) = nlim→∞tmax∈[a,b]|fm(t)−fn(t)|
= max
t∈[a,b]|nlim→∞fm(t)−nlim→∞fn(t)|
= max
t∈[a,b]|fm(t)−f(t)|
= d(fm, f) (m > n0)
❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♣❛r❛ t♦❞♦ t ∈[a, b]✱
|fm(t)−f(t)| ≤ε, (m > n0).
▼♦str❛♠♦s ❛ss✐♠ q✉❡ (fn(t)) ❝♦♥✈❡r❣❡ ✉♥✐❢♦r♠❡♠❡♥t❡ ❡♠ [a, b]✳ ❱✐st♦ q✉❡ ❛s
❢✉♥çõ❡s fn(t) sã♦ ❝♦♥tí♥✉❛s ❡♠ [a, b]✱ t❡♠♦s q✉❡ ❛ ❢✉♥çã♦ ❧✐♠✐t❡ f(t) é ❝♦♥tí♥✉❛
❡♠ [a, b] ✭✈❡❥❛ ❡♠ ❬✼❪✮✳ ❈♦♠ ✐ss♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ C[a, b] é ❝♦♠♣❧❡t♦✱ ❡♠ r❡❧❛çã♦ à
♠étr✐❝❛ ❞♦ ♠á①✐♠♦✳
✶✳✻ ❖ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤
◆❡st❡ tó♣✐❝♦ ✈❛♠♦s ❡♥✉♥❝✐❛r ❞❡♠♦♥str❛r ❡ ❡①✐❜✐r ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ❞♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤✳ ❯♠ ❞♦s ♠♦t✐✈♦s ❞❡ s✉❛ ✐♠❡♥s❛ ✐♠♣♦rtâ♥❝✐❛ é ♦ ❞❡ ❡❧❡ ❢♦r♥❡❝❡r ✉♠ ♣r♦❝❡ss♦ ✐t❡r❛t✐✈♦ ♣❛r❛ ❛ ❜✉s❝❛ ❞❡ s♦❧✉çõ❡s ❞❡ ❡q✉❛çõ❡s ♥ã♦ ❧✐♥❡❛r❡s✳ ❉❡✜♥✐çã♦ ✶✳✽ ❙❡❥❛(M, d)✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✳ ❯♠❛ ❢✉♥çã♦f :M →M é ❝❤❛♠❛❞❛ ❞❡ ❝♦♥tr❛çã♦ s♦❜r❡ ▼ s❡ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦ k <1✱ t❛❧ q✉❡✿
d(f(x), f(y))≤kd(x, y), ♣❛r❛ t♦❞♦ x, y ∈M.
❊①❡♠♣❧♦✿ ❈♦♥s✐❞❡r❡ M = R ❝♦♠ ❛ ♠étr✐❝❛ ✉s✉❛❧✳ ❆ ❢✉♥çã♦ f : [1,+∞) → R ❞❡✜♥✐❞❛ ♣♦r f(x) =√x é ✉♠❛ ❝♦♥tr❛çã♦✳ ❉❡ ❢❛t♦✿
d(f(x), f(y)) =|√x−√y|=|√x−√y| · |
√x+√y| |√x+√y| =
1
|√x+√y| · |x−y|. ❈♦♠♦ x, y ≥1✱ t❡♠♦s q✉❡ √x+√y ≥2✱ ♦✉ ❛✐♥❞❛✱ √ 1
x+√y ≤
1 2✳
▲♦❣♦✱
d(f(x), f(y))≤ 1
2|x−y|.
❖❜s❡r✈❡ q✉❡f ♥ã♦ é ✉♠❛ ❝♦♥tr❛çã♦ q✉❛♥❞♦ ❞❡✜♥✐❞❛ ♥♦ ✐♥t❡r✈❛❧♦ ❢❡❝❤❛❞♦[0,1]✱ ♣♦✐s lim
x→0+f
′
(x) = +∞✳
❚❡♦r❡♠❛ ✶✳✶ ✭❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤✮ ❈♦♥s✐❞❡r❡ (M, d) ✉♠ ❡s✲
♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦ ❡ ✉♠❛ ❝♦♥tr❛çã♦ f : M → M✳ ❊♥tã♦ f ♣♦ss✉✐ ✉♠ ú♥✐❝♦ ♣♦♥t♦ ✜①♦✳
Pr♦✈❛✿ ❈♦♥s✐❞❡r❡ x0 ∈ M ❡ ❛ s❡q✉ê♥❝✐❛ (xn) ❡♠ M ❞❡✜♥✐❞❛ ♣♦r xn+1 = f(xn)✳
❊♥tã♦✱
d(x1, x2) =d(f(x0), f(x1))≤kd(x0, x1)⇒d(x1, x2)≤kd(x0, x1)
❡
d(x2, x3) =d(f(x1), f(x2))≤kd(x1, x2)≤k2d(x0, x1)⇒d(x2, x3)≤k2d(x0, x1).
❈♦♥t✐♥✉❛♥❞♦ ♦ ♣r♦❝❡ss♦✱ ✉s❛♥❞♦ ✉♠ ❛r❣✉♠❡♥t♦ ✐♥❞✉t✐✈♦✱ ❝❤❡❣❛♠♦s à ❝♦♥❝❧✉sã♦ q✉❡ d(xn, xn+1)≤knd(x0, x1)✳ ❈♦♠♦ ♦ ♥♦ss♦ ✐♥t❡r❡ss❡ é ♠♦str❛r q✉❡(xn)é ✉♠❛ s❡q✉ê♥✲
❝✐❛ ❞❡ ❈❛✉❝❤②✱ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✱ t❡♠♦s✿
d(xn, xn+p)≤d(xn, xn+1) +d(xn+1, xn+2) +· · ·+d(xn+p−1, xn+p). ✭✶✳✻✮
P♦r ♦✉tr♦ ❧❛❞♦✱
d(xn, xn+1) ≤ knd(x0, x1),
d(xn+1, xn+2) ≤ kn+1d(x0, x1),
✳✳✳
d(xn+p−1, xn+p) ≤ kn+p−1d(x0, x1).
❆ss✐♠✱ t❡♠♦s✿
d(xn, xn+1)+d(xn+1, xn+2)+· · ·+d(xn+p−1, xn+p)≤ kn+kn+1+· · ·+kn+p−1
d(x0, x1).
❯s❛♥❞♦ ✭✶✳✻✮ ❡ q✉❡ k < 1✱ ♣❛r❛ q✉❛❧q✉❡r p✱ ✜①❛❞♦✱ t❡♠♦s
kn+kn+1+· · ·+kn+p−1 =kn· 1−k
p
1−k ≤ kn
1−k, ❧♦❣♦✱ ♦❜t❡♠♦s✿
d(xn, xn+p)≤
kn
1−k ·d(x0, x1). ❚♦♠❛♥❞♦ ♦ ❧✐♠✐t❡ q✉❛♥❞♦ n → ∞✱ ❝❤❡❣❛♠♦s ❛✿
lim
n→∞d(xn, xn+p)≤nlim→∞
kn
1−kd(x0, x1)
lim
n→∞d(xn, xn+p)≤d(x0, x1) limn→∞ kn
1−k. ❈♦♠♦ 0< k <1✱ kn
→0✱ ✈❛❧❡
lim
n→∞ kn
1−k = 0, ♦✉ s❡❥❛✱
lim
❞♦♥❞❡ ❝♦♥❝❧✉í♠♦s q✉❡ (xn) é ❞❡ ❢❛t♦ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ❡♠ M✳
❖r❛✱ ❝♦♠♦ (M, d) é ✉♠ ❊s♣❛ç♦ ▼étr✐❝♦ ❈♦♠♣❧❡t♦✱ ❡♥tã♦ (xn) ❝♦♥✈❡r❣❡ ❡♠ M✳
❆ss✐♠✱ t♦♠❛♥❞♦ ♦ ❧✐♠✐t❡ ♥❛ ❡q✉❛çã♦ xn+1 =f(xn) t❡r❡♠♦s✿
lim
n→∞xn+1 = limn→∞f(xn). ❇❡♠✱ ❝♦♠♦ lim
n→∞xn = a✱ ❡✱ ❝♦♠♦ ❛ ❛♣❧✐❝❛çã♦ f é ❝♦♥tí♥✉❛✱ ✉s❛♥❞♦ Pr♦♣♦s✐çã♦ ✺✱ ♦❜t❡♠♦s q✉❡
lim
n→∞f(xn) = f
lim
n→∞xn
=f(a). ❆ss✐♠✱ t❡♠♦s ❛ ✐❣✉❛❧❞❛❞❡ ❞❡s❡❥❛❞❛
f(a) = a.
❆ss✐♠✱ ♣r♦✈❛♠♦s ❛ ❡①✐stê♥❝✐❛ ❞♦ ♣♦♥t♦ ✜①♦✳ ❆❣♦r❛✱ ♣r♦✈❡♠♦s ❛ ✉♥✐❝✐❞❛❞❡✳ ❙❡❥❛♠ a ❡ b ❡♠ M t❛✐s q✉❡ f(a) =a ❡ f(b) =b✳ ❆ss✐♠✱
d(a, b) = d(f(a), f(b))≤kd(a, b). ■ss♦ ❧❡✈❛ à ❞❡s✐❣✉❛❧❞❛❞❡
(1−k)d(a, b)≤0.
❈♦♠♦ k < 1✱ ❡♥tã♦ 1−k < 0✱ ❞♦♥❞❡ ❝♦♥❝❧✉í♠♦s q✉❡ d(a, b) ≤ 0✳ ❈♦♠♦ d(a, b) é
✉♠ ♥ú♠❡r♦ r❡❛❧ ♥ã♦ ♥❡❣❛t✐✈♦✱ s❡❣✉❡ q✉❡ d(a, b) = 0✱ ❡ ✐ss♦ só ♦❝♦rr❡ s❡✱ ❡ s♦♠❡♥t❡
s❡✱ a=b✳
❆ss✐♠✱ f só ♣♦ss✉✐ ✉♠ ú♥✐❝♦ ♣♦♥t♦ ✜①♦✱ ♦ q✉❡ ❝♦♠♣❧❡t❛ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛✳
✶✳✼ ❆❧❣✉♠❛s ❆♣❧✐❝❛çõ❡s
❊①❡♠♣❧♦✿ ❈♦♥s✐❞❡r❡ ♦ ❊s♣❛ç♦ ▼étr✐❝♦ ❛ r❡t❛ r❡❛❧✱ ❝♦♠ ❛ ♠étr✐❝❛ ✉s✉❛❧✱ ❡ ❛ ❡q✉❛çã♦ ♥ã♦ ❧✐♥❡❛r
x=kcosx
♦♥❞❡ 0 < k < 1 é ✉♠❛ ❝♦♥st❛♥t❡ ❘❡❛❧✳ ❙❡rá q✉❡ ❡ss❛ ❡q✉❛çã♦ ♣♦ss✉✐ s♦❧✉çã♦❄ ❙❡
♣♦ss✉✐r✱ ❡ss❛ s♦❧✉çã♦ é ú♥✐❝❛❄
◆❛ ✐♥t❡♥çã♦ ❞❡ ✉s❛r ♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤✱ ❥á t❡♠♦s ✉♠ ❊s♣❛ç♦ ▼étr✐❝♦ ❈♦♠♣❧❡t♦ q✉❡ é ❛ r❡t❛ r❡❛❧✱ ❜❛st❛ ✈❡r✐✜❝❛r♠♦s s❡ f(x) = kcosx é ✉♠❛ ❝♦♥tr❛çã♦✳ ❱❡❥❛♠♦s q✉❡
d(f(x), f(y)) =k|cos(x)−cos(y)|=k
Z y
x
sen(t)dt
≤k
Z y
x |
❈♦♠♦ |sen(t)| ≤1 ✱ t❡♠♦s q✉❡
k
Z y
x |
sen(t)|dt ≤k|x−y|=kd(x, y). ❆ss✐♠✱
d(f(x), f(y))≤kd(x, y).
▲♦❣♦✱f é ✉♠❛ ❝♦♥tr❛çã♦ ❡ ♣❡❧♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤✱ ❛❞♠✐t❡ ✉♠ ú♥✐❝♦ ♣♦♥t♦ ✜①♦✱ ♦✉ s❡❥❛✱ ❛ ❡q✉❛çã♦ x=kcos(x)❛❞♠✐t❡ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦✳
❙❡❣✉✐♥❞♦ ❛s ✐t❡r❛çõ❡s ✐♥❞✐❝❛❞❛s ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛✱ ✐st♦ é✱ ♠♦♥t❛♥❞♦ ❛ s❡q✉ê♥❝✐❛ xn+1 = f(xn) ❡ t♦♠❛♥❞♦ k =
1
3✱ ♣♦r ❡①❡♠♣❧♦✱ ❡ x0 = 0,5 ❝♦♠♦ ✈❛❧♦r
✐♥✐❝✐❛❧✱ ♣♦❞❡♠♦s ❝❤❡❣❛r ❛♦s s❡❣✉✐♥t❡s ❝á❧❝✉❧♦s✿ x1 =
1
3cos(0,5) = 0,29252752
x2 =
1
3cos(0,29252752) = 0,31917269
x3 =
1
3cos(0,31917269) = 0,31649844
x4 =
1
3cos(0,31649844) = 0,31677702
x5 =
1
3cos(0,31677702) = 0,31674811
x6 =
1
3cos(0,31674811) = 0,31675111.
❖❜s❡r✈❡ q✉❡ ♥❛ q✉✐♥t❛ ✐t❡r❛çã♦ ❥á t❡♠♦s ✉♠❛ ♣r❡❝✐sã♦ ❞❡4❝❛s❛s ❞❡❝✐♠❛✐s s❡♥❞♦
❛ r❛✐③ ❞❛ ❡q✉❛çã♦ x≈0,3167 ✳ ❱❡❥❛ ❛ ✐❧✉str❛çã♦ ❣rá✜❝❛ ❛ s❡❣✉✐r✳
❊①❡♠♣❧♦✿ ❈♦♥s✐❞❡r❡ ♦ ❊s♣❛ç♦ ▼étr✐❝♦ ❛ r❡t❛ r❡❛❧✱ ❝♦♠ ❛ ♠étr✐❝❛ ✉s✉❛❧✱ ❡ ❛ ❡q✉❛çã♦ ♥ã♦ ❧✐♥❡❛r x= e−x✳ ❊st❛ ❡q✉❛çã♦ ♣♦ss✉✐ s♦❧✉çã♦❄ ❙❡ ♣♦ss✉✐r✱ s❡rá q✉❡ ❡❧❛
é ú♥✐❝❛❄
❇❡♠✱ s❡❥❛ f : [0,1] →[0,1] ❞❡✜♥✐❞❛ ♣♦r f(x) = e−x✳ ❱✐st♦ q✉❡ ♦ ✐♥t❡r✈❛❧♦ ]0,1]
é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦✱ ❞❡✈❡♠♦s ♠♦str❛r q✉❡ f é ✉♠❛ ❝♦♥tr❛çã♦✳ ❱❡❥❛♠♦s q✉❡ f′(x) =−e−x✱ ❡ t❡♠♦s q✉❡
|f′(x)
|=| −e−x
|=e−x <1,
s❡♠♣r❡ q✉❡ x∈ [0,1]✳ ❈♦♠♦ f′(x) é ❛ ✐♥❝❧✐♥❛çã♦ ❞❛ r❡t❛ ♥♦ ♣♦♥t♦(x, f(x))✱ t❡♠♦s q✉❡
f(x)−f(y)
x−y =f ′
(x)
f(x)−f(y)
x−y
=|f′(x)
|
|f(x)−f(y)|
|x−y| <1
|f(x)−f(y)|<|x−y|.
❱❡❥❛♠♦s ❝♦♠♦ ❣❛r❛♥t✐♠♦s ❣r❛✜❝❛♠❡♥t❡ ♦ ♣♦♥t♦ ✜①♦ ❞❛ ❢✉♥çã♦✳
❆ss✐♠✱ f é ❞❡ ❢❛t♦ ✉♠❛ ❝♦♥tr❛çã♦✱ ❡✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤✱ ❛ ❡q✉❛çã♦ ♥ã♦ ❧✐♥❡❛r x=e−x ♣♦ss✉✐ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ ♣❛r❛ x
∈]0,1]✳ ❆❣♦r❛✱ ✈❛♠♦s
❡st✐♠❛r ❡ss❛ s♦❧✉çã♦✳ ❈♦♠❡❝❡♠♦s t♦♠❛♥❞♦ x0 =
1
5 ✭❡st❡ ✈❛❧♦r é ❛❧❡❛tór✐♦ ❡ ❝♦❧❤✐❞♦
❡♠ ]0,1]✮✳
x1 =f
1 5
=e−15 = 0,818730753078
x2 =f(0,818730753078) =e−0,818730753078 = 0,440991025943
x3 =f(0,440991025943) =e−0,440991025943 = 0,643398480442
x4 =f(0,643398480442) =e−0,643398480442 = 0,525503472633
x5 =f(0,525503472633) =e−0,525503472633 = 0,591257607392
x6 =f(0,591257607392) =e−0,591257607392 = 0,553630596815
x7 =f(0,553630596815) =e−0,553630596815 = 0,574858936094
x8 =f(0,574858936094) =e−0,574858936094 = 0,562784251753
x9 =f(0,562784251753) =e−0,562784251753 = 0,569620885981
x10=f(0,569620885981) =e−0,569620885981 = 0,565739877967.
◆♦t❡♠♦s q✉❡ ❝♦♠ ❞❡③ ✐t❡r❛çõ❡s ❝❤❡❣❛♠♦s ❛♦ ✈❛❧♦r x = 0,56 ❝♦♠ ♣r❡❝✐sã♦ ❞❡
❞✉❛s ❝❛s❛s ❞❡❝✐♠❛✐s✳ ❆♣ós ✈ár✐❛s ✐t❡r❛çõ❡s ♣♦❞❡rí❛♠♦s ❝❤❡❣❛r ♣❡rt♦ ❞❛ s♦❧✉çã♦ q✉❡ é x = 0,56714329041✱ ♦♥❞❡ ❞❡♣❡♥❞❡♥❞♦ ❞❛ ♥❡❝❡ss✐❞❛❞❡ ❞❛ ❛♣r♦①✐♠❛çã♦ ♥ã♦ s❡r✐❛
♥❡❝❡ssár✐❛ ✉♠❛ ❣r❛♥❞❡ q✉❛♥t✐❞❛❞❡ ❞❡ ❝❛s❛s ❞❡❝✐♠❛✐s✳
❊①❡♠♣❧♦✿ ❖ ❝♦♥❤❡❝✐❞♦ ▼ét♦❞♦ ❞❡ ◆❡✇t♦♥ ❞❡ ❞❡t❡r♠✐♥❛çã♦ ❞❡ ③❡r♦s ❞❡ ❢✉♥çõ❡s r❡❛✐s ♣♦❞❡ s❡r ❡st✉❞❛❞♦ s♦❜ ❛ ✈✐sã♦ ❞♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤✳ ❙❡❥❛ f :R→R✉♠❛ ❢✉♥çã♦ ♥❛ q✉❛❧ ❞❡s❡❥❛♠♦s ❞❡t❡r♠✐♥❛r ✉♠ ③❡r♦✱ ♦✉ s❡❥❛✱ ✉♠❛ s♦❧✉çã♦ ♣❛r❛ ❞❛ ❡q✉❛çã♦✿
f(χ) = 0. ◆♦t❡ q✉❡ ❡ss❛ ❡q✉❛çã♦ ♣♦❞❡ s❡r r❡❡s❝r✐t❛ ♥❛ ❢♦r♠❛✿
χf′(χ) =χf′(χ)
−f(χ). P❛r❛ f′(χ)
6
= 0✱ t❡♠♦s✿
χ=χ− f(χ) f′(χ).
❈♦❧♦❝❛❞♦ ❞❡ss❛ ❢♦r♠❛✱ ♦ ♣r♦❜❧❡♠❛ t♦r♥❛✲s❡ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ♣♦♥t♦ ✜①♦ ♣❛r❛ ❛ ❛♣❧✐✲ ❝❛çã♦ T :R→R❞❡✜♥✐❞❛ ♣♦r
T(x) :=x− f(x) f′(x). ■ss♦ ♠♦t✐✈❛ ❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✳
Pr♦♣♦s✐çã♦ ✻ ❆ss✉♠❛ q✉❡ f : [a, b] → R é✱ ♣❡❧♦ ♠❡♥♦s✱ ❞✉❛s ✈❡③❡s ❞✐❢❡r❡♥❝✐á✈❡❧✱ ❝♦♠ f′(x)
6
= 0 ❡ f(a)f(b) < 0✱ ❡♥tã♦ f ♣♦ss✉✐rá ✉♠❛ r❛✐③ χ✱ ú♥✐❝❛✱ ❡♠ ✉♠ ❞❛❞♦ ✐♥t❡r✈❛❧♦ [a, b] s❡ ❡①✐st✐r λ ❝♦♠ 0< λ < 1 t❛❧ q✉❡
f(x)f′′(x)
[f′(x)]2
≤λ, ♣❛r❛ t♦❞♦ x∈[a, b] ✭✶✳✼✮
❡✱ s❡
f(x)
f′(x)
≤(1−λ)α, ✭✶✳✽✮
♦♥❞❡
x= a+b
2 ❡ α =
b−a
2 .
◆❡ss❡ ❝❛s♦✱ t❡♠✲s❡
χ= lim
n→∞xn,
♦♥❞❡ ❛ s❡q✉ê♥❝✐❛ (xn)∈[a, b] é ❞❡t❡r♠✐♥❛❞❛ ✐t❡r❛t✐✈❛♠❡♥t❡ ♣♦r
xn+1 =xn−
f(xn)
f′(x
n)
, n≥0, s❡♥❞♦ x0 ∈[a, b] ❛r❜✐trár✐♦✳
❖❜s❡r✈❛çã♦✿ ❈♦♠♦ ✈❡r❡♠♦s ❛ s❡❣✉✐r✱ ❛ ❝♦♥❞✐çã♦ ✭✶✳✼✮ é ✐♠♣♦rt❛♥t❡ ♣❛r❛ ❣❛r❛♥t✐r ❛ ❝♦♥tr❛t✐✈✐❞❛❞❡ ❞❡ T✱ ❡♥q✉❛♥t♦ q✉❡ ✭✶✳✽✮ é s✉✜❝✐❡♥t❡ ♣❛r❛ ❣❛r❛♥t✐r q✉❡ T ❧❡✈❡ ♣♦♥✲ t♦s ❞❡ [a, b] ❡♠ [a, b]✱ ♣♦❞❡♥❞♦ ❡✈❡♥t✉❛❧♠❡♥t❡ s❡r s✉❜st✐t✉í❞❛ ♣♦r ♦✉tr❛ ❝♦♥❞✐çã♦ q✉❡
❣❛r❛♥t❛ ♦ ♠❡s♠♦✳
Pr♦✈❛✿ ❙❡❥❛♠ x, y ∈[a, b]✳ ❚❡♠✲s❡
T(x)−T(y) = y− f(y)
f′(y)−x+ f(x)
f′(x)
= Z y x d dt
t− f(t) f′(t)
dt
=
Z y
x
f(t)f′′(t)
[f′(t)]2 dt. ❆ss✐♠✱ ✭✶✳✼✮ ❣❛r❛♥t❡ q✉❡✱
|T(x)−T(y)| =
Z y x
f(t)f′′(t)
[f′(t)]2 dt
≤ Z y x
f(t)f′′(t)
[f′(t)]2
dt ≤ Z y x
λ dt=λ|y−x|.
▲♦❣♦✱ ❝♦♠♦ |T(x)−T(y)| ≤λ|y−x|✱ ❝♦♠0≤λ <1✱ t❡♠♦s q✉❡T é ✉♠❛ ❝♦♥tr❛çã♦✳ Pr❡❝✐s❛♠♦s ❣❛r❛♥t✐r q✉❡ T ❧❡✈❡ ♣♦♥t♦s ❞❡ [a, b] ❡♠ [a, b]✳ ■ss♦ ❡q✉✐✈❛❧❡ ❛ ❣❛r❛♥t✐r
q✉❡ |T(x)−x| ≤ α✱ ♣❛r❛ t♦❞♦ x ∈ [a, b]✱ ♦✉ s❡❥❛✱ ♣❛r❛ t♦❞♦ x t❛❧ q✉❡ |x−x| ≤ α✳ ❯♠❛ ♠❛♥❡✐r❛ ❞❡ ✐♠♣♦r ✐ss♦ ✉s❛♥❞♦ ✭✶✳✼✮ é s✉♣♦r ✈á❧✐❞❛ ❛ ❝♦♥❞✐çã♦ ✭✶✳✽✮✳
