❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❈❛♠♣✉s ❞❡ ❘✐♦ ❈❧❛r♦
❯♠ ❡st✉❞♦ s♦❜r❡ s❡q✉ê♥❝✐❛s ❡ sér✐❡s
❆♥❛ ❈❡❝í❧✐❛ ❙❛♥❝❤❡s ❈❡rq✉❡✐r❛
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦✲ ♥❛❧ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡
❖r✐❡♥t❛❞♦r❛
Pr♦❢❛✳ ❉r❛✳ ❙✉③❡t❡ ▼❛r✐❛ ❙✐❧✈❛ ❆❢♦♥s♦
✺✶✵✳✵✼ ❈✹✶✻❡
❈❡rq✉❡✐r❛✱ ❆♥❛ ❈❡❝í❧✐❛ ❙❛♥❝❤❡s
❯♠ ❡st✉❞♦ s♦❜r❡ s❡q✉ê♥❝✐❛s ❡ sér✐❡s✴ ❆♥❛ ❈❡❝í❧✐❛ ❙❛♥❝❤❡s ❈❡rq✉❡✐r❛✲ ❘✐♦ ❈❧❛r♦✿ ❬s✳♥✳❪✱ ✷✵✶✸✳
✻✸ ❢✳✱ ✐❧✳✱ ❣rá❢s✳✱ t❛❜✳
❉✐ss❡rt❛çã♦ ✭♠❡str❛❞♦✮ ✲ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛✱ ■♥st✐✲ t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s✳
❖r✐❡♥t❛❞♦r❛✿ ❙✉③❡t❡ ▼❛r✐❛ ❙✐❧✈❛ ❆❢♦♥s♦
✶✳ ▼❛t❡♠át✐❝❛ ✲ ❊st✉❞♦ ❡ ❡♥s✐♥♦✳ ✷✳ ▲✐♠✐t❡✳ ✸✳ Pr♦❣r❡ssõ❡s✳ ✹✳ ❋✉♥çõ❡s✳ ■✳ ❚ít✉❧♦
❚❊❘▼❖ ❉❊ ❆P❘❖❱❆➬➹❖
❆♥❛ ❈❡❝í❧✐❛ ❙❛♥❝❤❡s ❈❡rq✉❡✐r❛
❯♠ ❡st✉❞♦ s♦❜r❡ s❡q✉ê♥❝✐❛s ❡ sér✐❡s
❉✐ss❡rt❛çã♦ ❛♣r♦✈❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ♥♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞♦ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑✱ ♣❡❧❛ s❡❣✉✐♥t❡ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✿
Pr♦❢❛✳ ❉r❛✳ ❙✉③❡t❡ ▼❛r✐❛ ❙✐❧✈❛ ❆❢♦♥s♦ ❖r✐❡♥t❛❞♦r❛
Pr♦❢❛✳ ❉r❛✳ ▲✐❣✐❛ ▲❛ís ❋ê♠✐♥❛
❋❛❝✉❧❞❛❞❡ ❞❡ ▼❛t❡♠át✐❝❛ ✴ ❯❋❯ ✲ ❯❜❡r❧â♥❞✐❛✴▼●
Pr♦❢❛✳ ❉r❛✳ ▼❛rt❛ ❈✐❧❡♥❡ ●❛❞♦tt✐ ■●❈❊✴ ❯♥❡s♣ ✲❘✐♦ ❈❧❛r♦✴ ❙P
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦✱ ♣r✐♠❡✐r❛♠❡♥t❡✱ ❛ ❉❡✉s✱ s❡♠ ♦ q✉❛❧ ♥❛❞❛ ❡♠ ♠✐♥❤❛ ✈✐❞❛ s❡r✐❛ ♣♦ssí✈❡❧✳ ❆❣r❛❞❡ç♦ ❛ t♦❞❛ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ❡♠ ❡s♣❡❝✐❛❧ ♠❡✉s ♣❛✐s✱ ▲✉✐③ ❏♦sé ❡ ▼❛r✐❛ ■♥ês✱ ♣❡❧♦ ❛♣♦✐♦ ❡ ❝♦♠♣r❡❡♥sã♦ ♥ã♦ s♦♠❡♥t❡ ❞✉r❛♥t❡ ❛ ❡❧❛❜♦r❛çã♦ ❞❡ss❛ ❞✐ss❡rt❛çã♦✱ ♠❛s ♣♦r s❡♠♣r❡ t❡r❡♠ s✐❞♦ ♠✐♥❤❛ ❜❛s❡ ❡ ❛❝r❡❞✐t❛r❡♠ ❡♠ ♠✐♠ ♠❡s♠♦ ♥♦s ♠♦♠❡♥t♦s ❡♠ q✉❡ ♥ã♦ ❛❝r❡❞✐t❡✐❀ ❡ ♠✐♥❤❛s ✐r♠ãs✱ ❆♥❛ P❛✉❧❛ ❡ ❆♥❛ ▲✉✐③❛✱ ♣❡❧❛ t♦r❝✐❞❛ ❡ ♣❡❧❛ ❛❥✉❞❛ q✉❛♥❞♦ ♣r❡❝✐s❡✐✳
❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛✱ Pr♦❢❛✳ ❉r❛✳ ❙✉③❡t❡ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛✱ ❝♦♥✜❛♥ç❛ ❡ ♦r✐❡♥t❛çã♦✱ s❡♠ ♦s q✉❛✐s ❡st❡ tr❛❜❛❧❤♦ ♥ã♦ t❡r✐❛ s✐❞♦ ❞❡s❡♥✈♦❧✈✐❞♦✳ ❚❛♠❜é♠ ❛❣r❛❞❡ç♦ ♣♦r s✉❛ ❛♠✐③❛❞❡ ❡ ♣❡❧❛s ♣❛❧❛✈r❛s ❞❡ â♥✐♠♦ ❛♦ ❧♦♥❣♦ ❞❡st❛ ❥♦r♥❛❞❛✳
❆❣r❛❞❡ç♦ ❛♦s ❛♠✐❣♦s q✉❡ ✜③ ♥♦ ❞❡❝♦rr❡r ❞♦ ▼❡str❛❞♦✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ▲✉❝✐❛♥♦✱ ●❧á✉✲ ❝✐❛✱ ▼❛r✐❛♥❛✱ ❙✐❜❡❧✐✱ P❛trí❝✐❛✱ ❈❛❧✐①t♦✱ ❘✐❝❛r❞♦✱ ❘♦❣ér✐♦ ❡ ❆r✐✳ ❱♦❝ês t♦r♥❛r❛♠ ❡ss❛ t❛r❡❢❛ ♠❡♥♦s ♣❡s❛❞❛ ❡ ♠❛✐s ❞✐✈❡rt✐❞❛✱ ♠❡s♠♦ ♥♦s ♠♦♠❡♥t♦s ❞❡ t❡♥sã♦ ❡ ❞❡ ❝❛♥s❛ç♦ ❢ís✐❝♦ ❡ ♠❡♥t❛❧✳
❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❛♠✐❣♦s ❞❡ ❧♦♥❣❛ ❞❛t❛✱ q✉❡ ❡♥t❡♥❞❡r❛♠ ♠✐♥❤❛ ❛✉sê♥❝✐❛ ❞✉r❛♥t❡ ❡ss❡ ♣❡rí♦❞♦ ❞❡ ❡st✉❞♦s ❡ ❢♦r❛♠ ✐♥❞✐s♣❡♥sá✈❡✐s ♣❛r❛ q✉❡ ❡✉ ❡♥❝♦♥tr❛ss❡ ❢♦rç❛ ❡ â♥✐♠♦✳
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦✱ ❛♣r❡s❡♥t❛♠♦s ❛ t❡♦r✐❛ ❞❡ s❡q✉ê♥❝✐❛s ❡ sér✐❡s ♥✉♠ér✐❝❛s✳ ❯♠❛ ✐♥✲ tr♦❞✉çã♦ ❛♦ ❡st✉❞♦ ❞❡ s❡q✉ê♥❝✐❛s ❡ sér✐❡s ❞❡ ❢✉♥çõ❡s r❡❛✐s t❛♠❜é♠ é ❛♣r❡s❡♥t❛❞❛✱ ❛ ✜♠ ❞❡ ❡①♣❧♦r❛r ❛s sér✐❡s ❞❡ ♣♦tê♥❝✐❛s ❡ ❛s sér✐❡s ❞❡ ❚❛②❧♦r✳ ❯s❛♥❞♦ ❛ t❡♦r✐❛ ❞❡s❡♥✲ ✈♦❧✈✐❞❛✱ ❛❜♦r❞❛♠♦s t❡♠❛s ♣r❡s❡♥t❡s ♥♦ ❈✉rrí❝✉❧♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❊❞✉❝❛çã♦ ❇ás✐❝❛✳ ❆❧é♠ ❞✐ss♦✱ ❛❧❣✉♠❛s s✉❣❡stõ❡s ❞❡ ♣r♦♣♦st❛s ❞✐❞át✐❝❛s sã♦ ❞❛❞❛s ❛♦s ♣r♦❢❡ss♦r❡s ❞❡ ▼❛t❡♠át✐❝❛ q✉❡ ❧❡❝✐♦♥❛♠ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳
❆❜str❛❝t
■♥ t❤✐s ✇♦r❦✱ ✇❡ ♣r❡s❡♥t t❤❡ t❤❡♦r② ♦❢ ♥✉♠❡r✐❝❛❧ s❡q✉❡♥❝❡s ❛♥❞ s❡r✐❡s✳ ❆♥ ✐♥tr♦❞✉❝✲ t✐♦♥ t♦ t❤❡ st✉❞② ♦❢ s❡q✉❡♥❝❡s ❛♥❞ s❡r✐❡s ♦❢ r❡❛❧ ❢✉♥❝t✐♦♥s ✐s ❛❧s♦ ♣r❡s❡♥t❡❞ ✐♥ ♦r❞❡r t♦ ❡①♣❧♦r❡ t❤❡ ♣♦✇❡r s❡r✐❡s ❛♥❞ ❚❛②❧♦r s❡r✐❡s✳ ❇② ✉s✐♥❣ t❤❡ t❤❡♦r② ❞❡✈❡❧♦♣❡❞✱ ✇❡ ❛♣♣r♦❛❝❤ t♦♣✐❝s ♣r❡s❡♥t ✐♥ t❤❡ ▼❛t❤❡♠❛t✐❝s ❈✉rr✐❝✉❧✉♠ ♦❢ ❇❛s✐❝ ❊❞✉❝❛t✐♦♥✳ ❋✉rt❤❡r♠♦r❡✱ s♦♠❡ s✉❣❣❡st✐♦♥s ♦❢ ❞✐❞❛❝t✐❝ ♣r♦♣♦s❛❧s ❛r❡ ❣✐✈❡♥ t♦ ♠❛t❤❡♠❛t✐❝s t❡❛❝❤❡rs ✇❤♦ t❡❛❝❤ ✐♥ ❍✐❣❤ ❙❝❤♦♦❧✳
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✶✺
✶ ❙❡q✉ê♥❝✐❛s ❡ sér✐❡s ♥✉♠ér✐❝❛s ✶✾
✶✳✶ ❙❡q✉ê♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✷ ▲✐♠✐t❡s ❞❡ s❡q✉ê♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✸ ❆r✐t♠ét✐❝❛ ❞♦s ❧✐♠✐t❡s ❞❡ s❡q✉ê♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✶✳✹ ❙ér✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✶✳✺ ❈♦♥✈❡r❣ê♥❝✐❛ ❡ ❞✐✈❡r❣ê♥❝✐❛ ❞❡ sér✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✶✳✻ ❚❡st❡s ❞❡ ❈♦♥✈❡r❣ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✶✳✼ ❙ér✐❡s ❛❧t❡r♥❛❞❛s ❡ ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥✈❡r❣❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✷ ■♥tr♦❞✉çã♦ às sér✐❡s ❞❡ ❢✉♥çõ❡s ✸✼
✷✳✶ ❙❡q✉ê♥❝✐❛s ❞❡ ❢✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✷ ❙ér✐❡s ❞❡ ❢✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✸ ❙ér✐❡s ❞❡ ♣♦tê♥❝✐❛s ❡ sér✐❡s ❞❡ ❚❛②❧♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
✸ ❙❡q✉ê♥❝✐❛s ❡ sér✐❡s ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦ ✹✼
✸✳✶ Pr♦❣r❡ssõ❡s ❛r✐t♠ét✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✶✳✶ Pr♦♣♦st❛ ❞❡ ❛t✐✈✐❞❛❞❡ ❞✐❞át✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✸✳✷ Pr♦❣r❡ssõ❡s ❣❡♦♠étr✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✸✳✷✳✶ Pr♦♣♦st❛s ❞❡ ❛t✐✈✐❞❛❞❡s ❞✐❞át✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✸✳✸ ❋✉♥çõ❡s ✈✐st❛s ❝♦♠♦ s♦♠❛s ✐♥✜♥✐t❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽
■♥tr♦❞✉çã♦
❆rq✉✐♠❡❞❡s ❞❡ ❙✐r❛❝✉s❛ ✭✷✽✼ ❛✳❈ ✲ ✷✶✷ ❛✳❈✮ é ❝♦♥s✐❞❡r❛❞♦ ✉♠ ❞♦s ♠❛✐♦r❡s ❝✐❡♥t✐st❛s ❞❛ ❆♥t✐❣✉✐❞❛❞❡ ❈❧áss✐❝❛✱ t❡♥❞♦ ❝♦♥tr✐❜✉í❞♦ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦✱ ❢ís✐❝♦✱ ❛str♦♥ô♠✐❝♦✳ ❊♠ ▼❛t❡♠át✐❝❛✱ ✉♠❛ ❞❡ s✉❛s ♠❛✐s ♥♦tá✈❡✐s ❝♦♥tr✐✲ ❜✉✐çõ❡s ❢♦✐ ♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛ ❞❡ ✉♠ ❝ír❝✉❧♦✱ ❛tr❛✈és ❞❡ s✉❝❡ss✐✈❛s ❛♣r♦①✐♠❛çõ❡s ❞❡st❛ ár❡❛ ♣♦r ♣♦❧í❣♦♥♦s ❝✉❥❛s ár❡❛s ❡r❛♠ ❝♦♥❤❡❝✐❞❛s❀ ♦ ♠ét♦❞♦ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r ❆rq✉✐♠❡❞❡s ♣❛r❛ ❡♥❝♦♥tr❛r ❛ ár❡❛ ❞❡ ✉♠ ❝ír❝✉❧♦ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ▼ét♦❞♦ ❞❛ ❊①❛✉stã♦✳
❆ ✐❞❡✐❛ ❡♠♣r❡❣❛❞❛ ♣♦r ❆rq✉✐♠❡❞❡s ♣♦❞❡ s❡r ✈✐s✉❛❧✐③❛❞❛ ♥❛ s❡q✉ê♥❝✐❛ ❞❡ ♣♦❧í❣♦♥♦s ✐♥s❝r✐t♦s ♥✉♠ ❝ír❝✉❧♦ ❞❡ r❛✐♦ r✱ ❛♣r❡s❡♥t❛❞❛ ♥❛ ✜❣✉r❛ ❛❜❛✐①♦✿
❋✐❣✉r❛ ✶✿ ▼ét♦❞♦ ❞❛ ❊①❛✉stã♦
P❛r❡❝❡ q✉❡ q✉❛♥t♦ ♠❛✐♦r ❢♦r ♦ ♥ú♠❡r♦ ❞❡ ❧❛❞♦s ❞♦ ♣♦❧í❣♦♥♦ ✐♥s❝r✐t♦ ♥❛ ❝✐r❝✉♥✲ ❢❡rê♥❝✐❛✱ ♠❛✐s ❛ ár❡❛ ❞❡st❡ ❡ ❞♦ ❝ír❝✉❧♦ s❡ ❛♣r♦①✐♠❛♠✱ ❞❡ ♠❛♥❡✐r❛ q✉❡✱ s❡ n ❝r❡s❝❡r
s❡♠ ❧✐♠✐t❡✱ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❛s ❞✉❛s ár❡❛s ❞❡✈❡rá s❡ t♦r♥❛r ✐♥s✐❣♥✐✜❝❛♥t❡ ❡✱ ❛ss✐♠✱ ♣♦❞❡r❡♠♦s ❝♦♥s✐❞❡r❛r ❛s ❞✉❛s ár❡❛s ❝♦♠♦ s❡♥❞♦ ✐❣✉❛✐s✳
❆♣❡s❛r ❞❡ ✐♥t✉✐t✐✈❛s ❛s ❛✜r♠❛çõ❡s ❢❡✐t❛s ♥♦ ♣❛rá❣r❛❢♦ ❛♥t❡r✐♦r✱ ✉s❛♠♦s ❛q✉✐ ❞✉❛s ✐❞❡✐❛s q✉❡ ♣r❡❝✐s❛♠ s❡r ❡①♣❧♦r❛❞❛s ❞❡ ♠❛♥❡✐r❛ ♣r❡❝✐s❛ ❡ r✐❣♦r♦s❛✿ ❛ ✐❞❡✐❛ ❞❡ q✉❡ ♦ ❧❛❞♦ ❞♦ ♣♦❧í❣♦♥♦ ✐♥s❝r✐t♦ ❡stá ❝r❡s❝❡♥❞♦ s❡♠ ❧✐♠✐t❡ ✭♦✉ ♥❛ ❧✐♥❣✉❛❣❡♠ ✉s✉❛❧✱n ❡stá t❡♥❞❡♥❞♦
❛♦ ✐♥✜♥✐t♦✮ ❡ ❛ ✐❞❡✐❛ ❞❡ q✉❡ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❛s ár❡❛s ❡stá s❡ t♦r♥❛♥❞♦ ♥✉❧❛ ✭♦✉ s❡❥❛✱ ✉♠ ✈❛❧♦r ❡stá s❡♥❞♦ ✧❛♣r♦①✐♠❛❞♦✧ ♣♦r ✉♠❛ s✉❝❡ssã♦ ❞❡ ♦✉tr♦s✮✳
✶✻
❋✐❜♦♥❛❝❝✐ ✭♣♦r ♠✉✐t♦s ❝♦♥s✐❞❡r❛❞♦ ♦ ♣r✐♠❡✐r♦ ❣r❛♥❞❡ ♠❛t❡♠át✐❝♦ ❡✉r♦♣❡✉ ❡ ✉♠ ❞♦s ♠❛✐s t❛❧❡♥t♦s♦s ❞❛ ■❞❛❞❡ ▼é❞✐❛✮✳ ❊♠ s❡✉ ❧✐✈r♦ ▲✐❜❡r ❆❜❛❝✐ ✭tr❛❞✉③✐❞♦ ♣❛r❛ ♦ ♣♦rt✉❣✉ês ❝♦♠♦ ▲✐✈r♦ ❞♦ ➪❜❛❝♦ ♦✉ ▲✐✈r♦ ❞♦ ❈á❧❝✉❧♦✮✱ ✉♠ ❞♦s ♣r♦❜❧❡♠❛s ❛❜♦r❞❛❞♦s r❡❢❡r❡✲s❡ à r❡♣r♦❞✉çã♦ ❞❡ ❝♦❡❧❤♦s ❡♠ ❝♦♥❞✐çõ❡s ♣ré✲❞❡t❡r♠✐♥❛❞❛s✳ ◆❛ ❧✐♥❣✉❛❣❡♠ ❛t✉❛❧✱ ❡st❡ ♣r♦❜❧❡♠❛ ♣♦❞❡ s❡r ❡♥✉♥❝✐❛❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
◆✉♠ ♣át✐♦ ❢❡❝❤❛❞♦✱ ❝♦❧♦❝❛✲s❡ ✉♠ ❝❛s❛❧ ❞❡ ❝♦❡❧❤♦s✳ ❙✉♣♦♥❞♦ q✉❡✱ ❡♠ ❝❛❞❛ ♠ês✱ ❛ ♣❛rt✐r ❞♦ s❡❣✉♥❞♦ ♠ês ❞❡ ✈✐❞❛✱ ❝❛❞❛ ❝❛s❛❧ ❞á ♦r✐❣❡♠ ❛ ✉♠ ♥♦✈♦ ❝❛s❛❧ ❞❡ ❝♦❡❧❤♦s✱ ❛♦ ✜♥❛❧ ❞❡ ✉♠ ❛♥♦✱ q✉❛♥t♦s ❝❛s❛✐s ❞❡ ❝♦❡❧❤♦s ❤❛✈❡rã♦ ♥♦ ♣át✐♦❄✳
❆ r❡s♦❧✉çã♦ ❞❡st❡ ♣r♦❜❧❡♠❛ é ❜❛st❛♥t❡ s✐♠♣❧❡s✳ ❖ t❡♠♣♦ t = 1 ✐♥❞✐❝❛ ♦ ♠♦♠❡♥t♦
❡♠ q✉❡ ♦ ♣r✐♠❡✐r♦ ❝❛s❛❧ ❞❡ ❝♦❡❧❤♦s ♥❛s❝❡✉ ❡ ❢♦✐ ❝♦❧♦❝❛❞♦ ♥♦ ♣át✐♦ ❢❡❝❤❛❞♦✳
❯♠ ♠ês ❞❡♣♦✐s ✭t = 2✮ ♦ ❝❛s❛❧ ❛✐♥❞❛ ♥ã♦ é ❢ért✐❧✱ ❞❡ ♠♦❞♦ q✉❡ ❛✐♥❞❛ ❡①✐st✐rá ❛♣❡♥❛s
✉♠ ❝❛s❛❧❀ ❝♦♥t✉❞♦✱ ❞❡❝♦rr✐❞♦ ♠❛✐s ✉♠ ♠ês(t= 3✮✱ ❡st❡ ❝❛s❛❧ ❥á ❡st❛rá ❢ért✐❧ ❡ ❤❛✈❡rã♦
❛❣♦r❛ ❞♦✐s ❝❛s❛✐s ❞❡ ❝♦❡❧❤♦s✳
◆♦ ♠ês s❡❣✉✐♥t❡ ✭t = 4✮✱ ♦ ❝❛s❛❧ ♥❛s❝✐❞♦ ♥♦ ♠ês ❛♥t❡r✐♦r ❛✐♥❞❛ ♥ã♦ é ❢ért✐❧✱ ♠❛s ♦
❝❛s❛❧ ♦r✐❣✐♥❛❧ ❣❡r❛rá ✉♠ ♥♦✈♦ ❝❛s❛❧ ❞❡ ✜❧❤♦t❡s❀ ♣♦rt❛♥t♦ ❛❣♦r❛ ❤❛✈❡rã♦ três ❝❛s❛✐s✳ ❉❡❝♦rr✐❞♦ ♠❛✐s ✉♠ ♠ês (t = 5)✱ ♦s ❞♦✐s ♣r✐♠❡✐r♦s ❝❛s❛✐s ❣❡r❛rã♦ ❞♦✐s ♥♦✈♦s ❝❛s❛✐s
❡✱ ❡♥tã♦✱ ❤❛✈❡rá ✉♠ t♦t❛❧ ❞❡ ❝✐♥❝♦ ❝❛s❛✐s ❞❡ ❝♦❡❧❤♦s✳ ➱ ❢á❝✐❧ ♣❡r❝❡❜❡r q✉❡✱ ❝❛❞❛ ❡❧❡✲
❋✐❣✉r❛ ✷✿ ❘❡♣r♦❞✉çã♦ ❞❡ ❈♦❡❧❤♦s ❡ ❙❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐
♠❡♥t♦ ❞❡st❛ s❡q✉ê♥❝✐❛✱ ❛ ♣❛rt✐r ❞♦ t❡r❝❡✐r♦✱ é ✐❣✉❛❧ ❛ s♦♠❛ ❞♦s ❞♦✐s ❛♥t❡r✐♦r❡s✳ ❆ss✐♠✱ ♣r♦ss❡❣✉✐♥❞♦ ♥❛ ❝♦♥str✉çã♦ ❞❛ s❡q✉ê♥❝✐❛ ❡♥❝♦♥tr❛♠♦s ❝♦♠♦ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ 144
❝❛s❛✐s ❞❡ ❝♦❡❧❤♦s✿
1,1,2,3,5,8,13,21,34,55,89,144, . . . ✭✶✮
■♥tr♦❞✉çã♦ ✶✼
❡s♣❡❝✐❛❧ é ♦ ❢❛t♦ ❞❡ q✉❡ ❡❧❛ ❛♣❛r❡❝❡ ♥ã♦ só ♥♦ ❡st✉❞♦ ❞❛ r❡♣r♦❞✉çã♦ ❞♦s ❝♦❡❧❤♦s✱ ♠❛s t❛♠❜é♠ ❡♠ ✐♥ú♠❡r♦s ❢❡♥ô♠❡♥♦s ♥❛t✉r❛✐s✳
P♦r ❡①❡♠♣❧♦✱ ❛❧❣✉♠❛s ♣❧❛♥t❛s ✭❝♦♠♦ ❛ ❆❝❤✐❧❧❡❛ Pt❛r♠✐❝❛✮ ♠♦str❛♠ ❡st❛ s❡q✉ê♥❝✐❛ ♥♦ ❝r❡s❝✐♠❡♥t♦ ❞❡ s❡✉s ❣❛❧❤♦s✳ ❖❜s❡r✈❡ ❛❜❛✐①♦✿
❋✐❣✉r❛ ✸✿ ❙❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ♥♦s ❣❛❧❤♦s ❞❡ ❆❝❤✐❧❧❡❛ Pt❛r♠✐❝❛
❉✐s❝✉t✐♠♦s ❛❝✐♠❛ ❞♦✐s ❡①❡♠♣❧♦s ❞❡ s❡q✉ê♥❝✐❛s✿ ❛ ♣r✐♠❡✐r❛ ❞❡❧❛s✱ ❣❡r❛❞❛ ♣❡❧❛ ❛♣r♦✲ ①✐♠❛çã♦ ❞❛ ár❡❛ ❞♦ ❝ír❝✉❧♦ ♣❡❧❛s ár❡❛s ❞❡ ♣♦❧í❣♦♥♦s r❡❣✉❧❛r❡s ❞❡ ❧❛❞♦ n✱ ❡ ❛ s❡❣✉♥❞❛
♦r✐❣✐♥❛❞❛ ❡♠ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ r❡♣r♦❞✉çã♦ ❞❡ ❝♦❡❧❤♦s✳
❯♠❛ s❡q✉ê♥❝✐❛ ✭♦✉ s✉❝❡ssã♦✮ é ✉♠❛ ❢✉♥çã♦a : N → R ❝✉❥♦ ❞♦♠í♥✐♦ é ♦ ❝♦♥❥✉♥t♦
❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❡ q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ✉♠ ❞❡st❡s ♥ú♠❡r♦s ♥❛t✉r❛✐s ✉♠ ♥ú♠❡r♦ r❡❛❧
an✳
❆❧❣✉♥s t✐♣♦s ❞❡ s❡q✉ê♥❝✐❛s ❝♦st✉♠❛♠ s❡r ❡st✉❞❛❞❛s ♥♦ ❊♥s✐♥♦ ❇ás✐❝♦✱ ❡s♣❡❝✐❛❧✲ ♠❡♥t❡ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✿ sã♦ ❛s ❝❤❛♠❛❞❛s ♣r♦❣r❡ssõ❡s ❛r✐t♠ét✐❝❛s ❡ ❣❡♦♠étr✐❝❛s✳ ❯♠❛ s❡q✉ê♥❝✐❛(an)s❡rá ❞❡♥♦♠✐♥❛❞❛ ♣r♦❣r❡ssã♦ ❛r✐t♠ét✐❝❛ s❡ ❝❛❞❛ t❡r♠♦✱ ❛ ♣❛rt✐r
❞♦ s❡❣✉♥❞♦✱ ❢♦r ♦❜t✐❞♦ ♣❡❧❛ s♦♠❛ ❞♦ t❡r♠♦ ❛♥t❡r✐♦r ❝♦♠ ✉♠❛ ❝♦♥st❛♥t❡r ∈R✱ ♦✉ s❡❥❛✱
an+1 =an+r, n≥1.
❯♠❛ s❡q✉ê♥❝✐❛(an)s❡rá ❞❡♥♦♠✐♥❛❞❛ ♣r♦❣r❡ssã♦ ❣❡♦♠étr✐❝❛ s❡ ❝❛❞❛ t❡r♠♦✱ ❛ ♣❛rt✐r
❞♦ s❡❣✉♥❞♦✱ ❢♦r ♦❜t✐❞♦ ♣❡❧♦ ♣r♦❞✉t♦ ❞♦ t❡r♠♦ ❛♥t❡r✐♦r ❝♦♠ ✉♠❛ ❝♦♥st❛♥t❡ q ∈R✱ ♦✉
s❡❥❛✱
an+1 =qan, n ≥1.
❚❛♠❜é♠ ♥♦ ❊♥s✐♥♦ ❇ás✐❝♦✱ sã♦ ❡st✉❞❛❞❛s ❛s ❞í③✐♠❛s ♣❡r✐ó❞✐❝❛s q✉❡ ♣♦❞❡♠ s❡r ✐♥t❡r♣r❡t❛❞❛s ❝♦♠♦ s♦♠❛s ✐♥✜♥✐t❛s✳
P♦r ❡①❡♠♣❧♦✱ ❝♦♥s✐❞❡r❡ ❛ r❡♣r❡s❡♥t❛çã♦ ❞❡❝✐♠❛❧ ❞❛ ❢r❛çã♦ 2
3✿ 2
✶✽
❡ ♦❜s❡r✈❡ q✉❡
0,6666· · ·= 6 10 +
6 100 +
6 1.000 +
6
10.000 +· · ·=
∞
X
n=1
6 10n.
❯♠❛ ✈❡③ q✉❡ ♥ã♦ ❢❛③ s❡♥t✐❞♦ s♦♠❛r ✉♠❛ q✉❛♥t✐❞❛❞❡ ✐♥✜♥✐t❛ ❞❡ t❡r♠♦s✱ é ♥❡❝❡ssár✐♦ ❡st❛❜❡❧❡❝❡r ✉♠❛ t❡♦r✐❛ ❝❛♣❛③ ❞❡ t♦r♥❛r ❛❝❡✐tá✈❡❧ ❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✳
❙❡♠♣r❡ q✉❡ t❡♥t❛r♠♦s s♦♠❛r ♦s t❡r♠♦s ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s (an)✱
♦❜t❡r❡♠♦s ✉♠❛ ❡①♣r❡ssã♦ ❞❛ ❢♦r♠❛ ∞
X
n=1
an =a1+a2+a3+· · ·+an+. . . .
➚ ❡st❛ s♦♠❛ ✐♥✜♥✐t❛ ❝❤❛♠❛♠♦s ❞❡ sér✐❡✳ P♦rt❛♥t♦✱ ❛ ❞í③✐♠❛ ♣❡r✐ó❞✐❝❛ ❛❝✐♠❛ ♣♦❞❡ s❡r ✐♥t❡r♣r❡t❛❞❛ ❝♦♠♦ ✉♠❛ sér✐❡ ♥✉♠ér✐❝❛✳
◆❡st❡ tr❛❜❛❧❤♦✱ ❛♣r❡s❡♥t❛♠♦s ❛ t❡♦r✐❛ ❞❡ s❡q✉ê♥❝✐❛s ❡ sér✐❡s ♥✉♠ér✐❝❛s ❡ ✐♥tr♦❞✉✲ ③✐♠♦s ❛ t❡♦r✐❛ ❞❡ s❡q✉ê♥❝✐❛s ❡ sér✐❡s ❞❡ ❢✉♥çõ❡s✳ ❆❧é♠ ❞✐ss♦✱ ❛♣r❡s❡♥t❛♠♦s ♣r♦♣♦st❛s ❞✐❞át✐❝❛s ❛♦s ♣r♦❢❡ss♦r❡s ❞❡ ♠❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ♣❛r❛ ♦ ❡♥s✐♥♦ ❞❡ t❡♠❛s r❡❧❛❝✐♦♥❛❞♦s às t❡♦r✐❛s ❛♣r❡s❡♥t❛❞❛s✳
✶ ❙❡q✉ê♥❝✐❛s ❡ sér✐❡s ♥✉♠ér✐❝❛s
❊st❡ ❝❛♣ít✉❧♦ ❞❡st✐♥❛r✲s❡✲á ❛♦ ❡st✉❞♦ ❞❡ s❡q✉ê♥❝✐❛s ❡ sér✐❡s ♥✉♠ér✐❝❛s✳ ❆q✉✐✱ ✈❡✲ r❡♠♦s ♦ ✐♠♣♦rt❛♥t❡ ❝♦♥❝❡✐t♦ ❞❡ ❧✐♠✐t❡ ❞❡ s❡q✉ê♥❝✐❛✱ ❜❡♠ ❝♦♠♦ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✱ ♦ q✉❡ ♥♦s ❛❥✉❞❛rá ❛ ❡st✐♣✉❧❛r ♦ ✈❛❧♦r ❞❡ ✉♠❛ s♦♠❛ ✐♥✜♥✐t❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s ✭t❛♠❜é♠ ❞❡♥♦♠✐♥❛❞❛ sér✐❡ ♥✉♠ér✐❝❛✮✳
✶✳✶ ❙❡q✉ê♥❝✐❛s
❉❡✜♥✐çã♦ ✶✳✶✳ ❯♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s é ✉♠❛ ❢✉♥çã♦ x : N → R q✉❡
❛ss♦❝✐❛ ❝❛❞❛ ♥ú♠❡r♦ ♥❛t✉r❛❧ n ❛ ✉♠ ♥ú♠❡r♦ r❡❛❧ x(n)✳ ❖ ✈❛❧♦r x(n)✱ ♣❛r❛ t♦❞♦ n∈N✱
s❡rá r❡♣r❡s❡♥t❛❞♦ ♣♦r xn ❡ ❞❡♥♦♠✐♥❛❞♦ ♥✲és✐♠♦ t❡r♠♦ ❞❛ s❡q✉ê♥❝✐❛✳
P♦r s✐♠♣❧✐❝✐❞❛❞❡✱ ❡s❝r❡✈❡♠♦s ❛♣❡♥❛s s❡q✉ê♥❝✐❛ ♥❡st❡ tr❛❜❛❧❤♦✱ ❞❡✈❡♥❞♦ ✜❝❛r s✉❜✲ ❡♥t❡♥❞✐❞♦ q✉❡ s❡ tr❛t❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s❀ ❛❧é♠ ❞✐ss♦✱ ❛❞♦t❛r❡♠♦s q✉❡ ♦ ♣r✐♠❡✐r♦ ❡❧❡✲ ♠❡♥t♦ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♥❛t✉r❛✐s é ♦ ♥ú♠❡r♦ 1✱ ♦ q✉❡ t♦r♥❛rá ♠❛✐s ❝ô♠♦❞❛ ❛ ♥♦t❛çã♦✳
❊s❝r❡✈❡r❡♠♦s(x1, x2, . . . , xn, . . .)✱ ♦✉ (xn)n∈N✱ ♦✉ s✐♠♣❧❡s♠❡♥t❡(xn)✱ ♣❛r❛ ✐♥❞✐❝❛r
❛ s❡q✉ê♥❝✐❛ x✳
➱ ❞❡ ❣r❛♥❞❡ ✈❛❧✐❛ r❡ss❛❧t❛r q✉❡ ♥ã♦ s❡ ♣♦❞❡ ❝♦♥❢✉♥❞✐r ❛ s❡q✉ê♥❝✐❛x❝♦♠ ♦ ❝♦♥❥✉♥t♦ x(N) ❞♦s s❡✉s t❡r♠♦s✳ P❛r❛ ❡ss❡ ❝♦♥❥✉♥t♦✱ ✉s❛r❡♠♦s ❛ ♥♦t❛çã♦{x1, x2, . . . , xn, . . .}✳ ❆
❢✉♥çã♦ x : N → R ♥ã♦ é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ✐♥❥❡t♦r❛✱ ♣♦❞❡✲s❡ t❡r xm = xn ❝♦♠ m 6= n✳
❖ ❝♦♥❥✉♥t♦ {x1, x2, . . . , xn, . . .} ✭❛♣❡s❛r ❞❛ ♥♦t❛çã♦✮ ♣♦❞❡ s❡r ✜♥✐t♦✱ ♦✉ ❛té ♠❡s♠♦
r❡❞✉③✐r✲s❡ ❛ ✉♠ ú♥✐❝♦ ❡❧❡♠❡♥t♦✱ ❝♦♠♦ é ♦ ❝❛s♦ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❝♦♥st❛♥t❡✱ ❡♠ q✉❡
xn =a ∈R ♣❛r❛ t♦❞♦ n∈N✳
❊①❡♠♣❧♦ ✶✳✶✳ ❈♦♥s✐❞❡r❡♠♦s ❛ s❡q✉ê♥❝✐❛ ❝✉❥♦ ♥✲és✐♠♦ t❡r♠♦ é xn =
1
n✳ ❊①♣❧✐❝✐t❛✲
♠❡♥t❡ ♣♦❞❡♠♦s ❡s❝r❡✈ê✲❧❛ ❝♦♠♦
1,1
2, 1 3,
1 4,
1 5, . . .
✳
◗✉❛❧q✉❡r t❡r♠♦ ❞❡st❛ s❡q✉ê♥❝✐❛ ❡stá ❧♦❝❛❧✐③❛❞♦ ♥♦ ✐♥t❡r✈❛❧♦ (0,1]✳ ❈♦♠ ❡❢❡✐t♦✱
❝♦♠♦ n ≥1≥0✱ ❡♥tã♦ xn=
1
n ≥0✳ P♦r ♦✉tr♦ ❧❛❞♦✱xn =
1
n ≤1✱ ♣❛r❛ t♦❞♦ n ≥1✳
❖❜s❡r✈❡ q✉❡ ♦s t❡r♠♦s ❞❡st❛ s❡q✉ê♥❝✐❛ ❡stã♦ ❞❡❝r❡s❝❡♥❞♦❀ ✐ss♦ é ❢á❝✐❧ ❞❡ s❡ ♣r♦✈❛r✿ ❝♦♠♦ n+ 1> n✱ s❡❣✉❡ q✉❡ 1
n+ 1 < 1
n✱ ♦✉ s❡❥❛✱xn+1 < xn✳
❆s ♦❜s❡r✈❛çõ❡s ❢❡✐t❛s ♥♦s ❞♦✐s ♣❛rá❣r❛❢♦s ❛♥t❡r✐♦r❡s ♣❡r♠✐t❡✲♥♦s ❛✜r♠❛r q✉❡ ❛ s❡q✉ê♥❝✐❛ (xn) =
1
n
é ❧✐♠✐t❛❞❛ ❡ ❞❡❝r❡s❝❡♥t❡✳
✷✵ ❙❡q✉ê♥❝✐❛s ❡ sér✐❡s ♥✉♠ér✐❝❛s
❊①❡♠♣❧♦ ✶✳✷✳ ❈♦♥s✐❞❡r❡♠♦s ❛ s❡q✉ê♥❝✐❛ ❝✉❥♦ n✲és✐♠♦ t❡r♠♦ é xn = n2✱ ♦♥❞❡ x1 =
1, x2 = 4, x3 = 9, . . .✱ ♦✉ s❡❥❛✱ ❛ s❡q✉ê♥❝✐❛ ❞♦s q✉❛❞r❛❞♦s ♣❡r❢❡✐t♦s✳ ➱ ❝❧❛r♦ q✉❡ ❡st❛ s❡q✉ê♥❝✐❛ é ❝r❡s❝❡♥t❡✱ ♣♦✐sxn+1 = (n+ 1)2 =n2+ 2n+ 1≥n2 =xn ❡ ♥♦t❡ q✉❡ ❡❧❛ ♥ã♦
♣♦ss✉✐ ❧✐♠✐t❛♥t❡✳
❆ss✐♠✱ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ ❡st❛ s❡q✉ê♥❝✐❛ é ✐❧✐♠✐t❛❞❛ ❡ ❝r❡s❝❡♥t❡✳ ❖s ❊①❡♠♣❧♦s ✶✳✶ ❡ ✶✳✷ ♠♦t✐✈❛♠ ❛s ❞❡✜♥✐çõ❡s q✉❡ s❡❣✉❡♠✳
❉❡✜♥✐çã♦ ✶✳✷✳ ❯♠❛ s❡q✉ê♥❝✐❛ (xn) s❡rá ❞✐t❛ ❧✐♠✐t❛❞❛ q✉❛♥❞♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s s❡✉s
t❡r♠♦s ❢♦r ❧✐♠✐t❛❞♦✱ ✐st♦ é✱ q✉❛♥❞♦ ❡①✐st✐r❡♠ ♥ú♠❡r♦s r❡❛✐s a ❡ b t❛✐s q✉❡ a ≤xn ≤ b
♣❛r❛ t♦❞♦ n∈N✳
◗✉❛♥❞♦ ✉♠❛ s❡q✉ê♥❝✐❛ ♥ã♦ ❢♦r ❧✐♠✐t❛❞❛✱ ❞✐r❡♠♦s q✉❡ ❡❧❛ é ✐❧✐♠✐t❛❞❛✳ ❆❧é♠ ❞✐ss♦✱ ✉♠❛ s❡q✉ê♥❝✐❛ s❡rá ❞✐t❛ ❧✐♠✐t❛❞❛ s✉♣❡r✐♦r♠❡♥t❡ q✉❛♥❞♦ ❡①✐st✐r ✉♠ ♥ú♠❡r♦ r❡❛❧
b t❛❧ q✉❡ xn ≤ b ♣❛r❛ t♦❞♦ n ∈ N✳ ◆❡st❡ ❝❛s♦✱ ❣❡♦♠❡tr✐❝❛♠❡♥t❡✱ ❛ s❡q✉ê♥❝✐❛ (xn)
❡st❛rá ❧♦❝❛❧✐③❛❞❛ ♥❛ s❡♠✐rr❡t❛(−∞, b]✳ ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✱ ❛ s❡q✉ê♥❝✐❛ s❡rá ❧✐♠✐t❛❞❛
✐♥❢❡r✐♦r♠❡♥t❡ q✉❛♥❞♦ ❡①✐st✐r ✉♠ r❡❛❧at❛❧ q✉❡a≤xn♣❛r❛ t♦❞♦n ∈N✳ ◆❡st❡ ❝❛s♦✱ ❛
s❡q✉ê♥❝✐❛(xn)❡st❛rá ❧♦❝❛❧✐③❛❞❛ ♥❛ s❡♠✐rr❡t❛ [a,+∞)✳ ❊✈✐❞❡♥t❡♠❡♥t❡✱ ✉♠❛ s❡q✉ê♥❝✐❛
s❡rá ❧✐♠✐t❛❞❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❢♦r ❧✐♠✐t❛❞❛ ✐♥❢❡r✐♦r ❡ s✉♣❡r✐♦r♠❡♥t❡✳ ◆ã♦ é ❞✐❢í❝✐❧ ✈❡r✐✜❝❛r q✉❡ s❡ (xn)é ❧✐♠✐t❛❞❛ ∃c > 0t❛❧ q✉❡ |xn|< c ✱∀n ∈N
◆♦ ❊①❡♠♣❧♦ ✶✳✶✱ ❛ s❡q✉ê♥❝✐❛ ❛♣r❡s❡♥t❛❞❛ é ❧✐♠✐t❛❞❛ ✐♥❢❡r✐♦r♠❡♥t❡ ♣♦r 0 ❡ s✉♣❡r✐✲
♦r♠❡♥t❡ ♣♦r 1✳ ❏á ♥♦ ❊①❡♠♣❧♦ ✶✳✷✱ ❛ s❡q✉ê♥❝✐❛ é ❛♣❡♥❛s ❧✐♠✐t❛❞❛ ✐♥❢❡r✐♦r♠❡♥t❡ ♣♦r 1✳
❉❡✜♥✐çã♦ ✶✳✸✳ ❯♠❛ s❡q✉ê♥❝✐❛ (xn) s❡rá ❞✐t❛ ❝r❡s❝❡♥t❡ q✉❛♥❞♦ xn< xn+1 ♣❛r❛ t♦❞♦
n∈N✳ ◗✉❛♥❞♦ xn≤xn+1 ♣❛r❛ t♦❞♦ n∈N✱ ❛ s❡q✉ê♥❝✐❛ s❡rá ❞✐t❛ ♥ã♦✲❞❡❝r❡s❝❡♥t❡✳
❆♥❛❧♦❣❛♠❡♥t❡✱ q✉❛♥❞♦ xn > xn+1 ♣❛r❛ t♦❞♦ n ∈ N✱ ❛ s❡q✉ê♥❝✐❛ s❡rá ❞✐t❛ ❞❡❝r❡s✲ ❝❡♥t❡✳ ❊❧❛ s❡rá ❞✐t❛ ♥ã♦✲❝r❡s❝❡♥t❡ q✉❛♥❞♦ xn ≥xn+1 ♣❛r❛ t♦❞♦n ∈N✳
❆s s❡q✉ê♥❝✐❛s ❝r❡s❝❡♥t❡s✱ ♥ã♦✲❞❡❝r❡s❝❡♥t❡s✱ ❞❡❝r❡s❝❡♥t❡s ❡ ♥ã♦✲❝r❡s❝❡♥t❡s sã♦ ❞❡✲ ♥♦♠✐♥❛❞❛s s❡q✉ê♥❝✐❛s ♠♦♥ót♦♥❛s✳
◗✉❛♥❞♦ r❡str✐♥❣✐r♠♦s ♦ ❞♦♠í♥✐♦ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ (xn) ❛ q✉❛❧q✉❡r s✉❜❝♦♥❥✉♥t♦
N′
⊂ N♦❜t❡r❡♠♦s ✉♠❛ ♥♦✈❛ s❡q✉ê♥❝✐❛ (x′
n) ❡ ❞✐r❡♠♦s q✉❡ (x
′
n) é ✉♠❛ s✉❜s❡q✉ê♥❝✐❛
❞❡(xn)✳
❊①❡♠♣❧♦ ✶✳✸✳ ◆♦ ❊①❡♠♣❧♦ ✶✳✷✱ ❝♦♥s✐❞❡r❛♠♦s ❛ s❡q✉ê♥❝✐❛(xn) = (n2)❞♦s q✉❛❞r❛❞♦s
♣❡r❢❡✐t♦s✳ ❚♦♠❛♥❞♦ N′
= {n′ |n′
= 2n−1, n ∈ N}✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ❛ r❡str✐çã♦
❞❡ (xn) ❛♦ ❝♦♥❥✉♥t♦ N′ ❡ ❡s❝r❡✈❡r ❛ s❡q✉ê♥❝✐❛ ❞♦s q✉❛❞r❛❞♦s ♣❡r❢❡✐t♦s í♠♣❛r❡s ❝✉❥♦
n✲és✐♠♦ t❡r♠♦ é x′
n = (2n−1)2✳ ➱ ❝❧❛r♦ q✉❡ (x
′
n) é ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❞❡ (xn)✳ ❉❡
♠♦❞♦ s❡♠❡❧❤❛♥t❡ ♣♦❞❡♠♦s ♦❜t❡r ❛ s❡q✉ê♥❝✐❛ ❞♦s q✉❛❞r❛❞♦s ♣❡r❢❡✐t♦s ♣❛r❡s t♦♠❛♥❞♦
N′′
={n′′ |n′′
= 2n, n∈N}✳
❯♠❛ q✉❡stã♦ q✉❡ s❡ ❝♦❧♦❝❛ ♥❛t✉r❛❧♠❡♥t❡ é ❛ ❞❡ s❛❜❡r s❡ ♦s t❡r♠♦s ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ✭♣❛r❛ í♥❞✐❝❡s s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡s✮ s❡ ❛♣r♦①✐♠❛♠ ❞❡ ✉♠ ♥ú♠❡r♦ r❡❛❧a ❡ ♣❡r♠❛♥❡✲
▲✐♠✐t❡s ❞❡ s❡q✉ê♥❝✐❛s ✷✶
✶✳✷ ▲✐♠✐t❡s ❞❡ s❡q✉ê♥❝✐❛s
◆❡ss❛ s❡çã♦✱ s❡rá ❝♦♥✈❡♥✐❡♥t❡ ♣❡♥s❛r ❡♠ ✉♠❛ s❡q✉ê♥❝✐❛(xn)❞❡ ♥ú♠❡r♦s r❡❛✐s ❝♦♠♦
✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♣♦♥t♦s ♥❛ r❡t❛ ✭❥á q✉❡ ❡①✐st❡ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ ♦s ♥ú♠❡r♦s r❡❛✐s ❡ ♦s ♣♦♥t♦s ❞❛ r❡t❛✮ ❡ ♥♦ s❡✉ ❧✐♠✐t❡ ✭q✉❡ ✈❛♠♦s ❞❡✜♥✐r ❛❜❛✐①♦✮ ❝♦♠♦ ✉♠ ♣♦♥t♦ ❞♦ q✉❛❧ ♦s ♣♦♥t♦s xn t♦r♥❛♠✲s❡ ❡ ♣❡r♠❛♥❡❝❡♠ ❛r❜✐tr❛r✐❛♠❡♥t❡ ♣ró①✐♠♦s✱
❞❡s❞❡ q✉❡ ♦ í♥❞✐❝❡n s❡❥❛ ❣r❛♥❞❡ ♦ s✉✜❝✐❡♥t❡✳ ❆ ✐♥t✉✐çã♦ ❣❡♦♠étr✐❝❛✱ ❛♣❡s❛r ❞❡ ♥ã♦ s❡r
s✉✜❝✐❡♥t❡ ♣❛r❛ ❡st❛❜❡❧❡❝❡r ❞❡ ♠❛♥❡✐r❛ r✐❣♦r♦s❛ ♦s r❡s✉❧t❛❞♦s✱ ♣♦❞❡ s❡r ♠✉✐t♦ út✐❧ ♣❛r❛ ❡♥t❡♥❞ê✲❧♦s✳
❉❡✜♥✐çã♦ ✶✳✹✳ ❖ ♥ú♠❡r♦ r❡❛❧ a s❡rá ❞✐t♦ ❧✐♠✐t❡ ❞❛ s❡q✉ê♥❝✐❛ (xn) ❞❡ ♥ú♠❡r♦s
r❡❛✐s s❡✱ ♣❛r❛ ❝❛❞❛ ǫ >0✱ ❢♦r ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ✉♠ ♥❛t✉r❛❧ n0 ∈N t❛❧ q✉❡ |xn−a|< ǫ
s❡♠♣r❡ q✉❡ n > n0✳ ❙✐♠❜♦❧✐❝❛♠❡♥t❡✿
limxn=a.≡.∀ǫ >0,∃n0 ∈N;n > n0 ⇒ |xn−a|< ǫ.
❊s❝r❡✈❡r❡♠♦s limxn =a✳ ❆s ♥♦t❛çõ❡s lim
n∈Nxn =a ❡ n→lim+∞xn = a t❛♠❜é♠ sã♦ ❡♥❝♦♥✲ tr❛❞❛s ♥❛ ❧✐t❡r❛t✉r❛✳
❖ sí♠❜♦❧♦ .≡. s✐❣♥✐✜❝❛ q✉❡ ♦ q✉❡ ✈❡♠ ❞❡♣♦✐s é ❞❡✜♥✐çã♦ ❞♦ q✉❡ ✈❡♠ ❛♥t❡s❀
∀s✐❣♥✐✜❝❛ ♣❛r❛ t♦❞♦❀ ∃s✐❣♥✐✜❝❛ ❡①✐st❡❀
;s✐❣♥✐✜❝❛ t❛❧ q✉❡❀
⇒s✐❣♥✐✜❝❛ ✐♠♣❧✐❝❛✳
➱ ✐♠♣♦rt❛♥t❡ ♦❜s❡r✈❛r q✉❡✿ limxn = a s❡✱ ❡ s♦♠❡♥t❡ s❡✱ limxn −a = 0✳ ❊st❛
❛✜r♠❛çã♦ s❡❣✉❡ ❞✐r❡t❛♠❡♥t❡ ❞❛ ❉❡✜♥✐çã♦ ✶✳✹✳
❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛✱ s❡ limxn =a✱ ❡♥tã♦ t♦❞♦ ✐♥t❡r✈❛❧♦(a−ǫ, a+ǫ)
❝♦♥t❡rá t♦❞♦s ♦s t❡r♠♦s ❞❛ s❡q✉ê♥❝✐❛ (xn) ❝♦♠ ❡①❝❡çã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡❧❡s✳
◗✉❛♥❞♦ ❡①✐st✐r ♦ ❧✐♠✐t❡ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❡ ❢♦r ✐❣✉❛❧ ❛ ✉♠ ♥ú♠❡r♦ r❡❛❧ a✱ ❞✐r❡♠♦s
q✉❡ ❡❧❛ é ❝♦♥✈❡r❣❡♥t❡ ❡ ❡s❝r❡✈❡r❡♠♦s xn → a✳ P♦❞❡♠♦s ❞✐③❡r t❛♠❜é♠ q✉❡ (xn)
❝♦♥✈❡r❣❡ ♣❛r❛ a ♦✉ q✉❡ (xn) t❡♥❞❡ ♣❛r❛ a✳ ❈❛s♦ ❝♦♥trár✐♦✱ ❞✐r❡♠♦s q✉❡ ❛ s❡q✉ê♥❝✐❛ é
❞✐✈❡r❣❡♥t❡✳
❊①❡♠♣❧♦ ✶✳✹✳ ◆♦✈❛♠❡♥t❡✱ ❝♦♥s✐❞❡r❡♠♦s ❛ s❡q✉ê♥❝✐❛ (xn) =
1
n
✳ ❱✐♠♦s✱ ♥♦ ❊①❡♠✲ ♣❧♦ ✶✳✶✱ q✉❡ ❡st❛ s❡q✉ê♥❝✐❛ é ❧✐♠✐t❛❞❛ ❡ ❞❡❝r❡s❝❡♥t❡✳ ◆♦ss❛ ✐♥t✉✐çã♦ ♥♦s ❞✐③ q✉❡ ❡st❛ s❡q✉ê♥❝✐❛ t❡♥❞❡ ❛ ③❡r♦✱ ✉♠❛ ✈❡③ q✉❡ ♦s t❡r♠♦s ✈ã♦ s❡ t♦r♥❛♥❞♦ ❝❛❞❛ ✈❡③ ♠❛✐s ♣ró✲ ①✐♠♦s ❞♦ ③❡r♦✱ s❡♠ ❝♦♥t✉❞♦ ✉❧tr❛♣❛ssá✲❧♦✳ Pr♦✈❡♠♦s✱ ♣♦✐s✱ ❛tr❛✈és ❞❛ ❞❡✜♥✐çã♦✱ q✉❡
limxn= 0✳
❉❡ ❢❛t♦✱ ❞❛❞♦ ǫ > 0 q✉❛❧q✉❡r✱ ♣♦❞❡♠♦s ♦❜t❡r n0 ∈ N t❛❧ q✉❡ n0 >
1
ǫ✳ ❊♥tã♦ n > n0 ⇒
1
n <
1
n0
< ǫ✱ ♦✉ s❡❥❛✱ n > n0 ⇒
1
n −0
< ǫ✳
❈♦♥s✐❞❡r❡ ❛❣♦r❛ q✉❡ ✉♠❛ s❡q✉ê♥❝✐❛ (xn) s❡❥❛ ❝♦♥✈❡r❣❡♥t❡✳ ❊♥tã♦✱ ❡❧❛ ♥ã♦ ♣♦❞❡
❝♦♥✈❡r❣✐r ♣❛r❛ ❞♦✐s ❧✐♠✐t❡s ❞✐st✐♥t♦s ❡✱ ❛❧é♠ ❞✐ss♦✱ t♦❞❛ s✉❜s❡q✉ê♥❝✐❛ (x′
✷✷ ❙❡q✉ê♥❝✐❛s ❡ sér✐❡s ♥✉♠ér✐❝❛s
❝♦♥✈❡r❣❡ ♣❛r❛ ♦ ♠❡s♠♦ ❧✐♠✐t❡ q✉❡ (xn)✳ ❊st❡s ❞♦✐s r❡s✉❧t❛❞♦s ❡st❛rã♦ ♣r❡s❡♥t❡s ♥♦s
♣ró①✐♠♦s t❡♦r❡♠❛s✳
❚❡♦r❡♠❛ ✶✳✶ ✭❯♥✐❝✐❞❛❞❡ ❞♦ ▲✐♠✐t❡✮✳ ❙❡❥❛(xn)✉♠❛ s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡✳ ❙❡limxn=
a ❡ limxn =b✱ ❡♥tã♦ a =b✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ limxn = a✳ ❉❛❞♦ q✉❛❧q✉❡r ♥ú♠❡r♦ r❡❛❧ b 6=a✱ ♠♦str❛r❡♠♦s q✉❡
♥ã♦ s❡ t❡♠ limxn = b✳ P❛r❛ ✐ss♦✱ t♦♠❡♠♦s ǫ = |
b−a|
2 > 0✳ P❛r❛ ❡st❡ ǫ✱ ❛✜r♠❛♠♦s
q✉❡ ♦s ✐♥t❡r✈❛❧♦s (a −ǫ, a +ǫ) ❡ (b −ǫ, b +ǫ) sã♦ ❞✐s❥✉♥t♦s✳ ❉❡ ❢❛t♦✱ s❡ ❡①✐st✐ss❡
x ∈ (a−ǫ, a+ǫ)∩(b−ǫ, b+ǫ)✱ t❡rí❛♠♦s |x−a| < ǫ ❡ |x−b| < ǫ✱ ❞♦♥❞❡ |a−b| ≤
|a−x|+|x−b|<2ǫ=|a−b|✱ ✉♠ ❛❜s✉r❞♦✳ ❈♦♠♦ limxn =a✱ ❡①✐st❡ n0 t❛❧ q✉❡✱ ♣❛r❛
n > n0✱ xn ∈ (a−ǫ, a+ǫ) ❡✱ ♣♦rt❛♥t♦✱ xn 6∈ (b−ǫ, b+ǫ)✱ ♣❛r❛ t♦❞♦ n > n0✳ ▲♦❣♦✱
limxn 6=b✳
❚❡♦r❡♠❛ ✶✳✷✳ ❙❡ limxn=a ❡♥tã♦ t♦❞❛ s✉❜s❡q✉ê♥❝✐❛ ❞❡ (xn) ❝♦♥✈❡r❣✐rá ♣❛r❛ ♦ ❧✐♠✐t❡
a✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ (x′
n) ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❞❡ (xn)✳ ❉❛❞♦ ǫ > 0✱ ❡①✐st❡ n0 ∈ N t❛❧ q✉❡ n > n0 ⇒ |xn−a| < ǫ✳ ❈♦♠♦ ♦s í♥❞✐❝❡s ❞❛ s✉❜s❡q✉ê♥❝✐❛ ❢♦r♠❛♠ ✉♠ ❝♦♥❥✉♥t♦
✐♥✜♥✐t♦✱ ❡①✐st❡ ❡♥tr❡ ❡❧❡s ✉♠ ni0 > n0✳ ❊♥tã♦✱ ❞❛❞♦ ni > ni0 s❡❣✉❡ q✉❡ ni > n0✱ q✉❡
✐♠♣❧✐❝❛|xni −a|< ǫ✳ ▲♦❣♦✱ limxni =a✳
❈♦♠♦ ❜❛s❡ ♥♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r✱ ♥♦t❡ q✉❡✿ s❡ limxn = a ❡♥tã♦✱ ♣❛r❛ t♦❞♦ k ∈ N✱
limxn+k =a✳ ❉❡ ❢❛t♦✱ (x1+k, x2+k, x3+k, . . . , xn+k, . . .) é ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❞❡ (xn)✳
❖ t❡♦r❡♠❛ ❛ s❡❣✉✐r ♥♦s ❢♦r♥❡❝❡ ❝♦♥❞✐çõ❡s s✉✜❝✐❡♥t❡s ♣❛r❛ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛✱ ♠❡s♠♦ q✉❡ ♦ s❡✉ ❧✐♠✐t❡ s❡❥❛ ❞❡s❝♦♥❤❡❝✐❞♦✳
❚❡♦r❡♠❛ ✶✳✸✳ ❚♦❞❛ s❡q✉ê♥❝✐❛ ♠♦♥ót♦♥❛ ❧✐♠✐t❛❞❛ é ❝♦♥✈❡r❣❡♥t❡✳
❉❡♠♦♥str❛çã♦✳ P❛r❛ ✜①❛r ✐❞❡✐❛s✱ t♦♠❡♠♦s (xn) ✉♠❛ s❡q✉ê♥❝✐❛ ♥ã♦✲❞❡❝r❡s❝❡♥t❡ ❧✐♠✐✲
t❛❞❛✳ ❈♦♥s✐❞❡r❡♠♦s a =✶sup{xn;n = 1,2,3, . . .}✳ ❉❡✈❡♠♦s t❡r a = limxn✳ ❉❡ ❢❛t♦✱
❞❛❞♦ q✉❛❧q✉❡r ǫ >0✱ ❝♦♠♦a−ǫ < a✱ ♦ ♥ú♠❡r♦a−ǫ ♥ã♦ é ❝♦t❛ s✉♣❡r✐♦r ❞♦ ❝♦♥❥✉♥t♦
❞♦s xn✳ P♦rt❛♥t♦✱ ❞❡✈❡ ❡①✐st✐r n0 ∈ N t❛❧ q✉❡ a−ǫ < xn0✳ ❈♦♠♦ ❛ s❡q✉ê♥❝✐❛ (xn) é
♠♦♥ót♦♥❛ ♥ã♦✲❞❡❝r❡s❝❡♥t❡✱ s❡ n > n0 ❡♥tã♦ xn0 ≤ xn ❡✱ ♣♦rt❛♥t♦✱ a−ǫ < xn✳ ❈♦♠♦
xn ≤ a✱ ✈❡♠♦s q✉❡ n > n0 ⇒ a−ǫ < xn < a+ǫ✳ P♦r ❝♦♥s❡❣✉✐♥t❡✱ limxn = a✱ ❝♦♠♦
q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳
❆ ❞❡♠♦♥str❛çã♦ ♥♦s ❝❛s♦s ❞❡ s❡q✉ê♥❝✐❛s ❞❡❝r❡s❝❡♥t❡s✱ ❝r❡s❝❡♥t❡s ❡ ♥ã♦✲❝r❡s❝❡♥t❡s é ❢❡✐t❛ ❞❡ ♠♦❞♦ ❛♥á❧♦❣♦✳
✶❯♠ ❡❧❡♠❡♥t♦ a ∈ Ré ❞❡♥♦♠✐♥❛❞♦ s✉♣r❡♠♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ X ⊂R q✉❛♥❞♦ a é ❛ ♠❡♥♦r ❞❛s
❝♦t❛s s✉♣❡r✐♦r❡s ❞❡ X ❡♠ R✳ ➚ ♣r♦♣ós✐t♦✱ ✉♠❛ ❝♦t❛ s✉♣❡r✐♦r ❞❡ X é ✉♠ ❡❧❡♠❡♥t♦ c ∈ R t❛❧ q✉❡
x≤c♣❛r❛ t♦❞♦x∈X✳ ❚♦❞♦ ❝♦♥❥✉♥t♦ ❧✐♠✐t❛❞♦ ♣♦ss✉✐ ✉♠ s✉♣r❡♠♦✳ P❛r❛ ♠❛✐♦r❡s ❞❡t❛❧❤❡s✱ ✈❡❥❛ ❬✻❪✱
▲✐♠✐t❡s ❞❡ s❡q✉ê♥❝✐❛s ✷✸
❖s ❚❡♦r❡♠❛s ✶✳✶ ❡ ✶✳✷ ♣♦❞❡♠ s❡r ❜❛st❛♥t❡ út❡✐s s❡ ❛♣❧✐❝❛❞♦s s✐♠✉❧t❛♥❡❛♠❡♥t❡✳ ❙❡ ❞❡s❡❥❛r♠♦s ♣r♦✈❛r q✉❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞✐✈❡r❣❡✱ ❜❛st❛ ❡①✐❜✐r ❞✉❛s s✉❜s❡q✉ê♥❝✐❛s q✉❡ ❝♦♥✈❡r❣❡♠ ♣❛r❛ ❧✐♠✐t❡s ❞✐❢❡r❡♥t❡s✳ ▼❛✐s ❛✐♥❞❛✱ s❡ q✉✐s❡r♠♦s ❞❡t❡r♠✐♥❛r ♦ ❧✐♠✐t❡ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ q✉❡ s❛❜❡♠♦s ❛ ♣r✐♦r✐ ❝♦♥✈❡r❣✐r✱ ❜❛st❛ ❞❡t❡r♠✐♥❛r♠♦s ♦ ❧✐♠✐t❡ ❞❡ q✉❛❧q✉❡r ✉♠❛ ❞❡ s✉❛s s✉❜s❡q✉ê♥❝✐❛s✳
❆♣❡s❛r ❞❡ ♦ ❚❡♦r❡♠❛ ✶✳✸ ♥♦s ❞❛r ✉♠ ❝r✐tér✐♦ ♣❛r❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ s❡q✉ê♥❝✐❛s✱ ❡❧❡ ♥ã♦ é ♦ ♠❛✐s ❣❡r❛❧ ♣♦ssí✈❡❧✱ ♣♦✐s ♠✉✐t❛s s❡q✉ê♥❝✐❛s sã♦ ❝♦♥✈❡r❣❡♥t❡s s❡♠ s❡r❡♠ ♠♦♥ó✲ t♦♥❛s✳ ❊①✐st❡ ✉♠ r❡s✉❧t❛❞♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❈r✐tér✐♦ ❞❡ ❈❛✉❝❤② q✉❡ ❢♦r♥❡❝❡ ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s ❡ s✉✜❝✐❡♥t❡s ♣❛r❛ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛✳
❉❡✜♥✐çã♦ ✶✳✺✳ ❯♠❛ s❡q✉ê♥❝✐❛(xn)❞❡ ♥ú♠❡r♦s r❡❛✐s s❡rá ❞✐t❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤②
s❡✱ ❞❛❞♦ ǫ > 0✱ ❢♦r ♣♦ssí✈❡❧ ♦❜t❡r n0 ∈ N t❛❧ q✉❡ m > n0 ❡ n > n0 ✐♠♣❧✐❝❛rã♦ |xm−xn|< ǫ✳
❖❜s❡r✈❡ q✉❡✱ ❛ ✜♠ ❞❡ q✉❡ ✉♠❛ s❡q✉ê♥❝✐❛ (xn) ❞❡ ♥ú♠❡r♦s r❡❛✐s s❡❥❛ ❞❡ ❈❛✉❝❤②✱ é
♥❡❝❡ssár✐♦ ❡ s✉✜❝✐❡♥t❡ q✉❡ s❡✉s t❡r♠♦s xm✱ xn s❡ ❛♣r♦①✐♠❡♠ ❛r❜✐tr❛r✐❛♠❡♥t❡ ✉♥s ❞♦s
♦✉tr♦s ♣❛r❛ í♥❞✐❝❡s s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡s✳ ◆♦t❡ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❡st❛ ❞❡✜♥✐çã♦ ❡ ❛ ❞❡✜♥✐çã♦ ❞❡ ❧✐♠✐t❡ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ♥❛ q✉❛❧ s❡ ❡①✐❣❡ q✉❡ ♦s t❡r♠♦s xn s❡ ❛♣r♦①✐♠❡♠
❛r❜✐tr❛r✐❛♠❡♥t❡ ❞❡ ✉♠ ✈❛❧♦r a✳ ◆❡ss❛ ♥♦✈❛ ❞❡✜♥✐çã♦ s❡ ✐♠♣õ❡ ✉♠❛ ❝♦♥❞✐çã♦ ❛♣❡♥❛s
s♦❜r❡ ♦s ♣ró♣r✐♦s t❡r♠♦s ❞❛ s❡q✉ê♥❝✐❛✳ ❊①❡♠♣❧♦ ✶✳✺✳ ❆ s❡q✉ê♥❝✐❛ (xn) =
1
n
é ❞❡ ❈❛✉❝❤②✳ ❉❡ ❢❛t♦✱ ❞❛❞♦ ǫ >0✱ ♣♦❞❡♠♦s
❡♥❝♦♥tr❛r n0 ∈ N t❛❧ q✉❡
1
n0
< ǫ✳ P♦rt❛♥t♦✱ ♣❛r❛ m, n≥ n0 ❡ s✉♣♦♥❞♦✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ q✉❡ n ≥m✱ t❡♠♦s 0< 1
n ≤
1
m ≤
1
n0✱ ❞❡ ♦♥❞❡ ❝♦♥❝❧✉í♠♦s q✉❡
1
n −
1
m
≤
1
n0 −
0
=
1
n0
< ǫ.
❖ ♣ró①✐♠♦ t❡♦r❡♠❛ ❝❛r❛❝t❡r✐③❛ t♦❞❛s ❛s s❡q✉ê♥❝✐❛s ❝♦♥✈❡r❣❡♥t❡s ❞❡ ♥ú♠❡r♦s r❡❛✐s ❡ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❈r✐tér✐♦ ❞❡ ❈❛✉❝❤② ♣❛r❛ s❡q✉ê♥❝✐❛s✳
❚❡♦r❡♠❛ ✶✳✹✳ ❯♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s s❡rá ❝♦♥✈❡r❣❡♥t❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❢♦r ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤②✳
P❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ❞❡st❡ t❡♦r❡♠❛ ♣r❡❝✐s❛r❡♠♦s ❞♦s r❡s✉❧t❛❞♦s ❡♥✉♥❝✐❛❞♦s ❛ s❡❣✉✐r✱ ❝✉❥❛s ❞❡♠♦♥str❛çõ❡s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ❡♠ ❬✻❪✱ ❈❛♣ít✉❧♦ ■❱✳
▲❡♠❛ ✶✳✶✳ ❚♦❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② é ❧✐♠✐t❛❞❛✳
▲❡♠❛ ✶✳✷✳ ❚♦❞❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s ♣♦ss✉✐ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❝♦♥✲ ✈❡r❣❡♥t❡✳
▲❡♠❛ ✶✳✸✳ ❙❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② (xn) ♣♦ss✉✐r ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ q✉❡ ❝♦♥✈❡r❣❡
✷✹ ❙❡q✉ê♥❝✐❛s ❡ sér✐❡s ♥✉♠ér✐❝❛s
❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✶✳✹✳ ❙❡❥❛ limxn = a✳ ❉❛❞♦ ǫ > 0✱ ❡①✐st❡ n0 ∈ N t❛❧ q✉❡
m > n0 ⇒ |xm−a| <
ǫ
2 ❡ n > n0 ⇒ |xn−a| <
ǫ
2✳ ▲♦❣♦✱ m, n > n0 ⇒ |xm−xn| ≤
|xm−a|+|xn−a|<
ǫ
2+
ǫ
2 =ǫ✱ ♦ q✉❡ ♠♦str❛ s❡r (xn) ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤②✳
❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ (xn) ❢♦r ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤②✱ ♣❡❧♦ ▲❡♠❛ ✶✳✶✱ ❡❧❛ s❡rá
❧✐♠✐t❛❞❛ ❡✱ ♣❡❧♦ ▲❡♠❛ ✶✳✷✱ ♣♦ss✉✐rá ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡✳ ❋✐♥❛❧♠❡♥t❡✱ ♣❡❧♦ ▲❡♠❛ ✶✳✸✱ (xn) s❡rá ❝♦♥✈❡r❣❡♥t❡✳
❖❜s❡r✈❡ q✉❡✱ ✐♥t✉✐t✐✈❛♠❡♥t❡✱ ♦ t❡♦r❡♠❛ ❛❝✐♠❛ ❛✜r♠❛ q✉❡ s❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡ ♣❛r❛ a ∈ R ❡♥tã♦ s❡✉s t❡r♠♦s ❛♦ s❡ ❛♣r♦①✐♠❛r❡♠ ❞♦ ♥ú♠❡r♦ r❡❛❧ a ♥❡❝❡ss❛r✐❛♠❡♥t❡
❛♣r♦①✐♠❛♠✲s❡ ✉♥s ❞♦s ♦✉tr♦s✳
❱❛♠♦s ✜♥❛❧✐③❛r ❡st❛ s❡çã♦ ❝♦♠ ♠❛✐s ✉♠ t❡♦r❡♠❛✱ q✉❡ ♣♦❞❡ s❡r ♦❜t✐❞♦ ❞✐r❡t❛♠❡♥t❡ ❞❛ ❞✐s❝✉ssã♦ ❛❝✐♠❛ ❛❝❡r❝❛ ❞❛s s❡q✉ê♥❝✐❛s ❞❡ ❈❛✉❝❤②❀ ❝♦♥t✉❞♦✱ ♦♣t❛♠♦s ♣♦r ❛♣r❡s❡♥t❛r ✉♠❛ ❞❡♠♦♥str❛çã♦ ❛❧t❡r♥❛t✐✈❛✳
❚❡♦r❡♠❛ ✶✳✺✳ ❚♦❞❛ s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡ é ❧✐♠✐t❛❞❛✳
❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❡♠♦s limxn = a✳ ❊♥tã♦✱ t♦♠❛♥❞♦ ǫ = 1✱ é ♣♦ssí✈❡❧ ❛✜r♠❛r
q✉❡ ❡①✐st❡ n0 ∈ N t❛❧ q✉❡ xn ∈ (a−1, a+ 1)✱ ♣❛r❛ n > n0✳ ❈♦♥s✐❞❡r❡♠♦s ♦ ❝♦♥❥✉♥t♦ ✜♥✐t♦ F ={x1, x2,· · · , xn0, a−1, a+ 1}✳ ❙❡❥❛♠ c❡ d ♦ ♠❡♥♦r ❡ ♦ ♠❛✐♦r ❡❧❡♠❡♥t♦ ❞❡
F✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙❡♥❞♦ ❛ss✐♠✱ t♦❞♦s ♦s t❡r♠♦s xn ❞❛ s❡q✉ê♥❝✐❛ ❡stã♦ ❝♦♥t✐❞♦s ♥♦
✐♥t❡r✈❛❧♦ [c, d]❡✱ ♣♦r ❝♦♥s❡❣✉✐♥t❡✱ ❛ s❡q✉ê♥❝✐❛ (xn) é ❧✐♠✐t❛❞❛✳
❖❜s❡r✈❛çã♦ ✶✳✶✳ ❆ r❡❝í♣r♦❝❛ ❞❡st❡ t❡♦r❡♠❛ é ❢❛❧s❛✳ ❆ s❡q✉ê♥❝✐❛ (0,7,0,7, . . .) é
❧✐♠✐t❛❞❛ ❡ ❞✐✈❡r❣❡♥t❡✳ P❛r❛ ❝♦♥st❛t❛r q✉❡ ❡❧❛ ❞✐✈❡r❣❡✱ ❜❛st❛ t♦♠❛r ❛s s✉❜s❡q✉ê♥❝✐❛s ❝♦♥st❛♥t❡s (x′
n) = (x2n−1) = (0) ❡(x′′n) = (x2n) = (7) ❡ ♦❜s❡r✈❛r q✉❡ x′n→0 ❡x
′′
n →7✳
P❡❧♦ ❚❡♦r❡♠❛ ✶✳✺✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡✱ s❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❢♦r ✐❧✐♠✐t❛❞❛✱ ❡♥tã♦ ❡❧❛ s❡rá ❞✐✈❡r❣❡♥t❡✳
❊♥tr❡ ❛s s❡q✉ê♥❝✐❛s ❞✐✈❡r❣❡♥t❡s✱ ✈❛♠♦s✱ ❛❣♦r❛✱ ❞❡st❛❝❛r ❛q✉❡❧❛s ❝✉❥♦s ✈❛❧♦r❡s s❡ t♦r♥❛♠ ❡ s❡ ♠❛♥tê♠ ❛r❜✐tr❛r✐❛♠❡♥t❡ ❣r❛♥❞❡s ♣♦s✐t✐✈❛♠❡♥t❡ ♦✉ ❛r❜✐tr❛r✐❛♠❡♥t❡ ❣r❛♥❞❡s ♥❡❣❛t✐✈❛♠❡♥t❡✳
❙❡❥❛ (xn) ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s✳ ❉✐r❡♠♦s q✉❡ ✧xn t❡♥❞❡ ♣❛r❛ ♠❛✐s
✐♥✜♥✐t♦✧✱ ❡ ❡s❝r❡✈❡♠♦s limxn = +∞ q✉❛♥❞♦✱ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ A ❞❛❞♦ ❛r❜✐tr❛✲
r✐❛♠❡♥t❡✱ ❢♦r ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r n0 ∈ N t❛❧ q✉❡ n > n0 ⇒ xn > A✱ ♦✉ s❡❥❛✱ ♣❛r❛
q✉❛❧q✉❡r A > 0 ❞❛❞♦✱ ❡①✐st✐rá ❛♣❡♥❛s ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ í♥❞✐❝❡s n t❛✐s q✉❡ xn ≤A✳
❆♥❛❧♦❣❛♠❡♥t❡✱ ❞✐r❡♠♦s q✉❡ ✧xnt❡♥❞❡ ♣❛r❛ ♠❡♥♦s ✐♥✜♥✐t♦✧✱ ❡ ❡s❝r❡✈❡♠♦slimxn =−∞
q✉❛♥❞♦✱ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ A ❞❛❞♦ ❛r❜✐tr❛r✐❛♠❡♥t❡✱ ❢♦r ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r n0 ∈N t❛❧ q✉❡ n > n0 ⇒ xn < −A✱ ♦✉ s❡❥❛✱ ♣❛r❛ q✉❛❧q✉❡r A > 0 ❞❛❞♦✱ ❡①✐st✐rá ❛♣❡♥❛s ✉♠
♥ú♠❡r♦ ✜♥✐t♦ ❞❡ í♥❞✐❝❡s n t❛✐s q✉❡ xn≤ −A✳
❱❡r✐✜❝❛✲s❡ ❢❛❝✐❧♠❡♥t❡ q✉❡ limn = +∞ ❡lim−n=−∞✳
❖❜s❡r✈❡ q✉❡ limxn = −∞ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ lim(−xn) = +∞✳ ◗✉❛♥❞♦ limxn =
+∞✱ ❛ s❡q✉ê♥❝✐❛(xn)é ✐❧✐♠✐t❛❞❛ s✉♣❡r✐♦r♠❡♥t❡✱ ♣♦ré♠ ❧✐♠✐t❛❞❛ ✐♥❢❡r✐♦r♠❡♥t❡✳ ◗✉❛♥❞♦
❆r✐t♠ét✐❝❛ ❞♦s ❧✐♠✐t❡s ❞❡ s❡q✉ê♥❝✐❛s ✷✺
❆ss✐♠✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ s❡q✉ê♥❝✐❛ ❝✉❥♦ t❡r♠♦ ❣❡r❛❧ éxn= (−1)nn♥ã♦ t❡♠+∞♥❡♠−∞
♣♦✐s é ✐❧✐♠✐t❛❞❛ ♥♦s ❞♦✐s s❡♥t✐❞♦s✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❛ s❡q✉ê♥❝✐❛(1,1,2,1,3,1,4,1,5,1, . . .)
é ❧✐♠✐t❛❞❛ ✐♥❢❡r✐♦r♠❡♥t❡✱ ✐❧✐♠✐t❛❞❛ s✉♣❡r✐♦r♠❡♥t❡✱ ♠❛s ♥ã♦ t❡♠ ❧✐♠✐t❡ ✐❣✉❛❧ ❛+∞♣♦r✲
q✉❡ ♣♦ss✉✐ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❝♦♥st❛♥t❡✳
✶✳✸ ❆r✐t♠ét✐❝❛ ❞♦s ❧✐♠✐t❡s ❞❡ s❡q✉ê♥❝✐❛s
◆❛ s❡çã♦ ❛♥t❡r✐♦r✱ ❡st❛❜❡❧❡❝❡♠♦s ❝♦♥❞✐çõ❡s s✉✜❝✐❡♥t❡s ✭s❡r ♠♦♥ót♦♥❛ ❡ ❧✐♠✐t❛❞❛✮ ♣❛r❛ q✉❡ ♦ ❧✐♠✐t❡ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❡①✐st❛✳ ◆❡st❛ s❡çã♦✱ ✐♥✈❡st✐❣❛r❡♠♦s ♦ ❝♦♠♣♦rt❛✲ ♠❡♥t♦ ❞❡st❡s ❧✐♠✐t❡s ❝♦♠ r❡❧❛çã♦ às ♦♣❡r❛çõ❡s ✉s✉❛✐s ❞❡ s♦♠❛✱ s✉❜tr❛çã♦✱ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ❞✐✈✐sã♦✳ ❖s ❞♦✐s t❡♦r❡♠❛s ❛q✉✐ ❛♣r❡s❡♥t❛❞♦s s❡rã♦ ✐❧✉str❛❞♦s ❝♦♠ ❡①❡♠♣❧♦s ♣❛r❛ ❢❛✲ ❝✐❧✐t❛r ♦ ❡♥t❡♥❞✐♠❡♥t♦✳
❚❡♦r❡♠❛ ✶✳✻✳ ❙❡limxn = 0❡(yn)❢♦r ✉♠❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛ ✭♠❡s♠♦ q✉❡ ❞✐✈❡r❣❡♥t❡✮
❡♥tã♦ limxnyn= 0✳
❉❡♠♦♥str❛çã♦✳ ❈♦♠♦ ❛ s❡q✉ê♥❝✐❛ (yn) é ❧✐♠✐t❛❞❛✱ ❡①✐st❡ c > 0 t❛❧ q✉❡ |yn| < c ♣❛r❛
t♦❞♦ n ∈ N✳ ❈♦♠♦ limxn = 0✱ ❞❛❞♦ ǫ > 0✱ ❡①✐st❡ n0 ∈ N t❛❧ q✉❡ n > n0 ⇒ |xn| < ǫ
c✳
❉❡st❡ ♠♦❞♦✱ ♣❛r❛ n > n0✱ t❡♠♦s✿
|xnyn|=|xn||yn|<
ǫ cc=ǫ
✱ ♦✉ s❡❥❛✱ limxnyn = 0✱ ❝♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳
❊①❡♠♣❧♦ ✶✳✻✳ ❉❛❞♦ x ∈ R✱ s❡❥❛ (xn) =
1
n
❡ (yn) = (sen(nx))✳ ◆♦t❡ q✉❡ xn → 0
✭❝♦♥❢♦r♠❡ ♦ ❊①❡♠♣❧♦ ✶✳✹✮ ❡ |yn| ≤1❀ ♣♦rt❛♥t♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✻✱ s❡❣✉❡ q✉❡
limsen(nx)
n = 0.
❖ t❡♦r❡♠❛ ❛ s❡❣✉✐r é r❡s♣♦♥sá✈❡❧ ♣♦r ✈❛❧✐❞❛r t♦❞❛ ❛ ♠❛♥✐♣✉❧❛çã♦ ❛r✐t♠ét✐❝❛ q✉❡ ♣♦❞❡♠♦s ❢❛③❡♠♦s ❝♦♠ ❞✉❛s ♦✉ ♠❛✐s s❡q✉ê♥❝✐❛s✳ ❊♠ ❧✐♥❤❛s ❣❡r❛✐s✱ ❡❧❡ ♥♦s ♠♦str❛ q✉❡ ❛s ♦♣❡r❛çõ❡s ❛r✐t♠ét✐❝❛s ✉s✉❛✐s sã♦ ♣r❡s❡r✈❛❞❛s ♣❡❧♦s ❧✐♠✐t❡s ❞❡ s❡q✉ê♥❝✐❛s✳
❚❡♦r❡♠❛ ✶✳✼✳ ❙❡❥❛♠ (xn) ❡ (yn) ❞✉❛s s❡q✉ê♥❝✐❛s ❝♦♥✈❡r❣❡♥t❡s✱ ❝♦♠ limxn = a ❡
limyn =b 6= 0✳ ❊♥tã♦✿
✶✳ lim(xn+yn) = a+b❀
✷✳ lim(xn−yn) =a−b❀
✸✳ lim(xnyn) =ab❀
✹✳ lim(cxn) = ca✱∀c∈R❀
✺✳ limxn
yn
= a
✷✻ ❙❡q✉ê♥❝✐❛s ❡ sér✐❡s ♥✉♠ér✐❝❛s
❉❡♠♦♥str❛çã♦✳ ✭✶✮ ❉❛❞♦ ǫ > 0✱ ❝♦♠♦ limxn = a ❡ limyn = b✱ ❡①✐st❡♠ ♥ú♠❡r♦s
♥❛t✉r❛✐sn1 ❡n2 t❛✐s q✉❡ n > n1 ⇒ |xn−a|<
ǫ
2 ❡n > n2 ⇒ |yn−b|<
ǫ
2✳ ❚♦♠❡
n0 = max{n1, n2}✳ ❊♥tã♦✱ ♣❛r❛ n > n0✱ t❡♠♦s✿
|(xn+yn)−(a+b)|=|(xn−a) + (yn−b)| ≤ |xn−a|+|yn−b|<
ǫ
2 +
ǫ
2 =ǫ.
✭✷✮ ❆ ♣r♦✈❛ ❞❡st❛ ♣r♦♣r✐❡❞❛❞❡ é ❛♥á❧♦❣❛ à ❛♥t❡r✐♦r✳ ✭✸✮ ❖❜s❡r✈❡ q✉❡
xnyn−ab=xnyn−xnb+xnb−ab=xn(yn−b) + (xn−a)b.
◆♦t❡ q✉❡ (xn) é ❧✐♠✐t❛❞❛ ♣♦rq✉❡ é ❝♦♥✈❡r❣❡♥t❡ ✭❝♦♥❢♦r♠❡ ♦ ❚❡♦r❡♠❛ ✶✳✺✮ ❡
lim(yn−b) = 0✳ ▲♦❣♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✻✱ lim[xn(yn−b)] = 0✳ ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱
lim[(xn−a)b] = 0❡✱ ♣❡❧♦ ✐t❡♠(1)✱ ❥á ❞❡♠♦♥str❛❞♦ ❛❝✐♠❛✱ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✱ ♣♦✐s
lim[xnyn−ab] = lim[xn(yn−b) + (xn−a)b]
= lim[xn(yn−b)] + lim[(xn−a)b]
= 0 + 0
= 0.
✭✹✮ ➱ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ ✐t❡♠ ❛♥t❡r✐♦r✱ t♦♠❛♥❞♦ yn =c✱ ∀n∈N✳
✭✺✮ ◆♦t❡♠♦s q✉❡✱ ❝♦♠♦ limynb =b2✱ ❡①✐st❡ n0 ∈ N t❛❧ q✉❡ n > n0 ⇒ynb >
b2
2✳ ❉❡
❢❛t♦✱ t♦♠❛♥❞♦ ǫ = b
2
2✱ ❡①✐st❡ n0 ∈ N t❛❧ q✉❡ n > n0 ⇒ |ynb−b
2| < b2
2✱ ❞♦♥❞❡
b2 −b2
2 < ynb < b
2+b2
2✱ ♣❛r❛ n > n0 ❡✱ ♣♦rt❛♥t♦✱ ynb >
b2
2 ♣❛r❛ n > n0✳ ❊♥tã♦✱
♣❛r❛ n > n0✱
1
ynb
é ✉♠ ♥ú♠❡r♦ ♣♦s✐t✐✈♦ ✐♥❢❡r✐♦r ❛ b2
2✳ ◆♦t❡ q✉❡
xn
yn −
a b =
bxn−ayn
ynb
= (bxn−ayn)
1
ynb
.
❈♦♠♦ lim(bxn−ayn) =ba−ab= 0✱ s❡❣✉❡ ❞♦ ❚❡♦r❡♠❛ ✶✳✻ q✉❡
limxn
yn −
a
b = 0 ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ lim xn
yn
= a
b.
❊①❡♠♣❧♦ ✶✳✼✳ ❈♦♥s✐❞❡r❡♠♦s ❛ s❡q✉ê♥❝✐❛ ❝✉❥♦ n✲és✐♠♦ t❡r♠♦ é xn = 6 +
e−n
n ✳ P❡❧♦
❚❡♦r❡♠❛ ✶✳✼✱ ✐t❡♥s (1) ❡ (3)✱ ❡ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✻✱ t❡♠♦s✿
lim(xn) = lim
6 + e
−n
n
= lim 6 + lim
e−n
n
= lim 6 + lime−nlim
1
n
= 6 + 0 = 6,
❙ér✐❡s ✷✼
❙❛❧✐❡♥t❛♠♦s q✉❡ ❞❡✈❡✲s❡ t♦♠❛r ♦ ❝✉✐❞❛❞♦ ❞❡ ♥ã♦ t❡♥t❛r ❛♣❧✐❝❛r ♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r ♣❛r❛ ❝❡rt❛s s♦♠❛s ✭♦✉ ♣r♦❞✉t♦s✮ ❡♠ q✉❡ ♦ ♥ú♠❡r♦ ❞❡ ♣❛r❝❡❧❛s ✭♦✉ ❢❛t♦r❡s✮ é ✈❛r✐á✈❡❧ ❡ ❝r❡s❝❡ ❛❝✐♠❛ ❞❡ q✉❛❧q✉❡r ❧✐♠✐t❡✳ ❖ ♣ró①✐♠♦ ❡①❡♠♣❧♦ ✐❧✉str❛ ❡ss❛ s✐t✉❛çã♦✳
❊①❡♠♣❧♦ ✶✳✽✳ ❈♦♥s✐❞❡r❡sn=
1
n +
1
n +
1
n+· · ·+
1
n ✉♠❛ s♦♠❛ ❝♦♠ n ♣❛r❝❡❧❛s✳ ❊♥tã♦ sn = 1 ❡✱ ♣♦rt❛♥t♦✱ limsn = 1✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❝❛❞❛ ♣❛r❝❡❧❛
1
n t❡♠ ❧✐♠✐t❡ ③❡r♦✳ ❯♠❛
❛♣❧✐❝❛çã♦ ❞❡s❝✉✐❞❛❞❛ ❞♦ ✐t❡♠(1) ❞♦ ❚❡♦r❡♠❛ ✶✳✼ ♥♦s ❝♦♥❞✉③✐r✐❛ ❛♦ ❛❜s✉r❞♦ ❞❡ ❝♦♥❝❧✉✐r
q✉❡
limsn = lim
1
n +
1
n +
1
n +· · ·+
1
n
= 0 + 0 + 0 +· · ·+ 0 = 0.
✶✳✹ ❙ér✐❡s
❙❡ t❡♥t❛r♠♦s s♦♠❛r ♦s t❡r♠♦s ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s(an)✱ ♦❜t❡r❡♠♦s
✉♠❛ ❡①♣r❡ssã♦ ❞❛ ❢♦r♠❛ ∞
X
n=1
an=a1+a2+a3+· · ·+an+. . . ✭✶✳✶✮
q✉❡ é ❞❡♥♦♠✐♥❛❞❛ sér✐❡ ✐♥✜♥✐t❛ ♦✉ s✐♠♣❧❡s♠❡♥t❡ sér✐❡✳ ◆♦ q✉❡ s❡❣✉❡✱ r❡♣r❡s❡♥t❛r❡♠♦s ✉♠❛ sér✐❡ t❛♠❜é♠ ♣♦r P
an ❡ ❝❤❛♠❛r❡♠♦s an ❞❡
t❡r♠♦ ❣❡r❛❧ ❞❛ sér✐❡✳
◗✉❛❧ é ♦ s✐❣♥✐✜❝❛❞♦ ❞❛ s♦♠❛ ✐♥✜♥✐t❛ ✭✶✳✶✮❄ ◆ã♦ ❢❛③ s❡♥t✐❞♦ s♦♠❛r ❝♦♥✈❡♥❝✐♦✲ ♥❛❧♠❡♥t❡ ✐♥✜♥✐t♦s t❡r♠♦s❀ ❛ ❢✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛ ♣❛r❛ t♦r♥❛r ❛❝❡✐tá✈❡❧ ✉♠❛ s♦♠❛ ✐♥✜♥✐t❛ é ♦ ❝♦♥❝❡✐t♦ ❞❡ ❧✐♠✐t❡ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛✱ ♦ q✉❛❧ ❥á ❞✐s❝✉t✐♠♦s ♥❛ s❡çã♦ ❛♥t❡r✐♦r✳ ❱❛♠♦s ❝♦♥s✐❞❡r❛r ❛ s❡q✉ê♥❝✐❛ ❢♦r♠❛❞❛ ♣❡❧❛s s♦♠❛s ♣❛r❝✐❛✐s ♦✉ r❡❞✉③✐❞❛s✱ q✉❡ é ❞❡✜♥✐❞❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
s1 =a1,
s2 =s1+a2 =a1+a2,
s3 =s2+a3 =a1 +a2+a3, ✳✳✳
sn =sn−1+xn=a1+a2+· · ·+an−1+an.
❊st❛s s♦♠❛s ♣❛r❝✐❛✐s ❢♦r♠❛♠ ✉♠❛ ♥♦✈❛ s❡q✉ê♥❝✐❛✱ ❞❡♥♦t❛❞❛ ♣♦r(sn)✱ q✉❡ ♣♦❞❡ ♦✉
♥ã♦ ❝♦♥✈❡r❣✐r✳
❙❡ ❡①✐st✐r ♦ ❧✐♠✐t❡
s= limsn = lim[a1+a2+· · ·+an−1+an],
❛ sér✐❡ P
an s❡rá ❝♦♥✈❡r❣❡♥t❡ ❡ ♦ ❧✐♠✐t❡ s s❡rá ❞❡♥♦♠✐♥❛❞♦ s♦♠❛ ❞❛ sér✐❡✳
✷✽ ❙❡q✉ê♥❝✐❛s ❡ sér✐❡s ♥✉♠ér✐❝❛s
❊①❡♠♣❧♦ ✶✳✾✳ ❈♦♥s✐❞❡r❡♠♦s ❛ sér✐❡ X 1
n(n+ 1)✳ ❱❛♠♦s ✐♥✈❡st✐❣❛r✱ ♥❛ s❡q✉ê♥❝✐❛ ❞❡
✐❞❡✐❛s ❛♣r❡s❡♥t❛❞❛s ❛❝✐♠❛✱ q✉❛❧ ❛ s♦♠❛ ❞❡st❛ sér✐❡ ✭s❡ é q✉❡ ❡❧❛✱ ❞❡ ❢❛t♦✱ ❡①✐st❡✮✳ ❱❛♠♦s ❝♦♥s✐❞❡r❛r ❛ s♦♠❛ ✐♥✜♥✐t❛ 1
2+ 1 6+
1
12+. . .✱ ❞❡ ♦♥❞❡ t❡♠♦s✿
s1 =
1 2,
s2 =s1+a2 =
1 2+
1 6 =
2 3,
s3 =
2
3+a3 = 2 3+
1 12 =
3 4,
✳✳✳
sn=
n n+ 1.
P❛r❛ ❞❡t❡r♠✐♥❛r ❛ ❡①♣r❡ssã♦ ❞❡ sn✱ ✉s❛♠♦s ♦ Pr✐♥❝í♣✐♦ ❞❡ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛
✭✈❡❥❛ ❬✻❪✱ ❈❛♣ít✉❧♦ ■■✮✳
❆ss✐♠✱ s♦♠♦s ❧❡✈❛❞♦s ❛ ❝♦♥s✐❞❡r❛r ❛ s❡q✉ê♥❝✐❛ (sn) =
n n+ 1
✱ q✉❡ ❝♦♥✈❡r❣❡ ♣❛r❛ ♦ ♥ú♠❡r♦ r❡❛❧ 1✳ ❊s❝r❡✈❡♠♦s✱ ♣♦rt❛♥t♦✱ X 1
n(n+ 1) = 1✳
✶✳✺ ❈♦♥✈❡r❣ê♥❝✐❛ ❡ ❞✐✈❡r❣ê♥❝✐❛ ❞❡ sér✐❡s
❋♦r♠❛❧✐③❛♥❞♦ ❛ ❞✐s❝✉ssã♦ ❛❝✐♠❛✱ t❡♠♦s ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✿ ❉❡✜♥✐çã♦ ✶✳✻✳ ❉❛❞❛ ✉♠❛ sér✐❡ P
an=a1+a2+a3+· · ·+an+. . .✱ ❞❡♥♦t❡♠♦s ♣♦r
sn s✉❛ ♥✲és✐♠❛ s♦♠❛ ♣❛r❝✐❛❧✿
sn= n
X
j=1
aj =a1+a2+a3 +· · ·+an.
❙❡ ❛ s❡q✉ê♥❝✐❛ (sn) ❢♦r ❝♦♥✈❡r❣❡♥t❡ ❡ limsn = s✱ ❡♥tã♦ ❛ sér✐❡ Pan s❡rá ❞✐t❛ ❝♦♥✲
✈❡r❣❡♥t❡ ❡ ❡s❝r❡✈❡r❡♠♦s P
an=s✳ ❖ ♥ú♠❡r♦ s é ❞❡♥♦♠✐♥❛❞♦ s♦♠❛ ❞❛ sér✐❡✳ ❈❛s♦
❝♦♥trár✐♦✱ ❛ sér✐❡ s❡rá ❞✐t❛ ❞✐✈❡r❣❡♥t❡✳
❖❜s❡r✈❡✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛✱ q✉❡ ❛ s♦♠❛ ❞❡ ✉♠❛ sér✐❡ é ♦ ❧✐♠✐t❡ ❞❛ s❡q✉ê♥❝✐❛ ❞❡ s✉❛s s♦♠❛s ♣❛r❝✐❛✐s✳ ❉❡s❞❡ ♠♦❞♦✱ q✉❛♥❞♦ ❡s❝r❡✈❡♠♦s P
an = s q✉❡r❡♠♦s ❞✐③❡r
q✉❡✱ s♦♠❛♥❞♦ ✉♠❛ q✉❛♥t✐❞❛❞❡ s✉✜❝✐❡♥t❡ ❞❡ t❡r♠♦s ❞❛ sér✐❡✱ ♣♦❞❡♠♦s ❝❤❡❣❛r tã♦ ♣❡rt♦ q✉❛♥t♦ q✉✐s❡r♠♦s ❞♦ ♥ú♠❡r♦s✳
❉❛s ♣r♦♣r✐❡❞❛❞❡s ❛r✐t♠ét✐❝❛s ❞♦ ❧✐♠✐t❡ ❞❡ s❡q✉ê♥❝✐❛s✱ ❝♦♥❝❧✉í♠♦s q✉❡ s❡ P
an
❡ P
bn ❢♦r❡♠ sér✐❡s ❝♦♥✈❡r❣❡♥t❡s✱ ❡♥tã♦ ❛ sér✐❡ P(an + bn) s❡rá ❝♦♥✈❡r❣❡♥t❡✱ ❝♦♠
P
(an+bn) = Pan+Pbn✳ ❆❧é♠ ❞✐ss♦✱ s❡ Pan ❝♦♥✈❡r❣✐r ❡♥tã♦✱ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦
r❡❛❧ M✱ t❡r✲s❡✲á P
❈♦♥✈❡r❣ê♥❝✐❛ ❡ ❞✐✈❡r❣ê♥❝✐❛ ❞❡ sér✐❡s ✷✾
❖❜s❡r✈❛çã♦ ✶✳✷✳ ❆ sér✐❡ ∞
X
n=1
an s❡rá ❝♦♥✈❡r❣❡♥t❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ sér✐❡
∞
X
n=n0
an ❢♦r
❝♦♥✈❡r❣❡♥t❡✱ ♦♥❞❡ n0 é ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ❛r❜✐trár✐♦✳ ❉❡ ❢❛t♦✱ s❡ ❛s r❡❞✉③✐❞❛s ❞❛ ♣r✐♠❡✐r❛ sér✐❡ ❢♦r❡♠ sn✱ ❛s ❞❛ s❡❣✉♥❞❛ s❡rã♦ tn+1 =sn+n0 −sn0−1✱ ❝♦♠ t1 =an0✳ ❈♦♠
✐ss♦✱ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ ✉♠❛ sér✐❡ ♥ã♦ s❡ ❛❧t❡r❛ s❡ ❞❡❧❛ ♦♠✐t✐♠♦s ✭♦✉ ❛ ❡❧❛ ❛❝r❡s❝❡♥t❛♠♦s✮ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ t❡r♠♦s✳
❆ q✉❡stã♦ q✉❡ ♥❛t✉r❛❧♠❡♥t❡ s❡ ❛♣r❡s❡♥t❛✱ ❛ss✐♠ ❝♦♠♦ ♥♦ ❝❛s♦ ❞❛s s❡q✉ê♥❝✐❛s✱ é ❛ ❞❡ ❡♥t❡♥❞❡r q✉❛♥❞♦ ✉♠❛ sér✐❡ ❝♦♥✈❡r❣❡✳ ❊①✐st❡♠ t❡st❡s ♣❛r❛ ✈❡r✐✜❝❛r ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ♦✉ ♥ã♦ ❞❡ ✉♠❛ sér✐❡ ❡ ♥ós ♦s ❛♣r❡s❡♥t❛r❡♠♦s ♥❛ ♣ró①✐♠❛ s❡çã♦✳ ❆♥t❡s✱ ♦❜s❡r✈❛♠♦s q✉❡ ❛ ♣r✐♠❡✐r❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ♣❛r❛ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ ✉♠❛ sér✐❡ é q✉❡ ♦ s❡✉ t❡r♠♦ ❣❡r❛❧ ❝♦♥✈✐r❥❛ ♣❛r❛ ③❡r♦✳
❚❡♦r❡♠❛ ✶✳✽✳ ❙❡ P
an ❢♦r ✉♠❛ sér✐❡ ❝♦♥✈❡r❣❡♥t❡✱ ❡♥tã♦ liman= 0✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ sn = a1 +a2 +· · ·+an ❛ n✲és✐♠❛ s♦♠❛ ♣❛r❝✐❛❧ ❞❛ sér✐❡
P
an✳
P♦r ❤✐♣ót❡s❡✱ ❡①✐st❡ limsn✳ ❉✐❣❛♠♦s q✉❡ limsn = s✳ ➱ ❝❧❛r♦ q✉❡ t❛♠❜é♠ ❞❡✈❡♠♦s
t❡r limsn−1 = s ✉♠❛ ✈❡③ q✉❡ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ♥ã♦ é ❛❢❡t❛❞❛ q✉❛♥❞♦ r❡t✐r❛♠♦s ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ t❡r♠♦s ❞❡❧❛✳ P♦rt❛♥t♦✱
liman= lim(sn−sn−1) = limsn−limsn−1 =s−s= 0.
❆ r❡❝í♣r♦❝❛ ❞♦ ❚❡♦r❡♠❛ ✶✳✽ é ❢❛❧s❛✳ ❖ ❝♦♥tr❛✲❡①❡♠♣❧♦ ❝❧áss✐❝♦ é ❞❛❞♦ ♣❡❧❛ sér✐❡ ❤❛r♠ô♥✐❝❛ X1
n✳ ❙❡✉ t❡r♠♦ ❣❡r❛❧ xn =
1
n t❡♥❞❡ ♣❛r❛ ③❡r♦ ✭✈❡❥❛ ❊①❡♠♣❧♦ ✶✳✹✮✱ ♠❛s ❛
sér✐❡ ❞✐✈❡r❣❡✳ ❉❡ ❢❛t♦✱ t❡♠♦s
s2n = 1 +
1 2 + 1 3+ 1 4 + 1 5 + 1 6 + 1 7 + 1 8 +. . . 1
2n−1 + 1 +· · ·+
1 2n
>1 + 1 2 +
2 4 +
4
8 +· · ·+ 2n−1
2n = 1 +n
1 2.
❊♥tã♦✱ ❛ s❡q✉ê♥❝✐❛ (s2n)é ✐❧✐♠✐t❛❞❛ ❡✱ ♣♦rt❛♥t♦✱ ❛ sér✐❡
X 1
n é ❞✐✈❡r❣❡♥t❡✳
❈♦♥✈é♠ ♥♦t❛r q✉❡ ✉♠❛ sér✐❡ P
an ♣♦❞❡ ❞✐✈❡r❣✐r ♣♦r ❞♦✐s ♠♦t✐✈♦s✳ ❖✉ ♣♦rq✉❡ ❛
s❡q✉ê♥❝✐❛ ❞❛s r❡❞✉③✐❞❛s sn ♥ã♦ é ❧✐♠✐t❛❞❛ ♦✉ ♣♦rq✉❡ ❡❧❛s ♦s❝✐❧❛♠ ❡♠ t♦r♥♦ ❞❡ ❛❧❣✉♥s
✈❛❧♦r❡s✳ ◗✉❛♥❞♦ ♦s t❡r♠♦s ❞❛ sér✐❡ tê♠ t♦❞♦s ♦ ♠❡s♠♦ s✐♥❛❧✱ ❡st❛ ú❧t✐♠❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ♥ã♦ ❛❝♦♥t❡❝❡✱ ♣♦✐s ♥❡st❡ ❝❛s♦✱ ❛s r❡❞✉③✐❞❛s ❢♦r♠❛♠ ✉♠❛ s❡q✉ê♥❝✐❛ ♠♦♥ót♦♥❛✳ ❙❡❣✉❡ ✐♠❡❞✐❛t❛♠❡♥t❡ ❞♦ ❚❡♦r❡♠❛ ✶✳✸ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿
❚❡♦r❡♠❛ ✶✳✾✳ ❙❡❥❛ (an) ✉♠❛ s❡q✉ê♥❝✐❛ t❛❧ q✉❡an≥0 ♣❛r❛ t♦❞♦n ∈N✳ ❆ sér✐❡
P
an
❝♦♥✈❡r❣✐rá s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛s r❡❞✉③✐❞❛s sn ❢♦r♠❛r❡♠ ✉♠❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛✱ ✐st♦ é✱
