❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛
❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚
❊①✐stê♥❝✐❛ ❡ ❯♥✐❝✐❞❛❞❡ ❞♦s
◆ú♠❡r♦s ❘❡❛✐s ✈✐❛ ❈♦rt❡s ❞❡
❉❡❞❡❦✐♥❞✳
†♣♦r
❑❡r❧② ▼♦♥r♦❡ P♦♥t❡s
s♦❜ ♦r✐❡♥t❛çã♦ ❞❡
Pr♦❢✳❉r ◆❛♣♦❧❡ó♥ ❈❛r♦ ❚✉❡st❛
❉✐ss❡rt❛çã♦ ❛♣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♦ ❞❡
P814e Pontes, Kerly Monroe.
Existência e unicidade dos números reais via Corte de Dedekind / Kerly Monroe Pontes.-- João Pessoa, 2014. 75f.
Orientador: Napoleón Caro Tuesta Dissertação (Mestrado) - UFPB/CCEN
1. Matemática. 2. Cortes de Dedekind. 3. Corpo ordenado completo. 4. Isomorfismo.
❊①✐stê♥❝✐❛ ❡ ❯♥✐❝✐❞❛❞❡ ❞♦s
◆ú♠❡r♦s ❘❡❛✐s ✈✐❛ ❈♦rt❡s ❞❡
❉❡❞❡❦✐♥❞✳
♣♦r
❑❡r❧② ▼♦♥r♦❡ P♦♥t❡s
❉✐ss❡rt❛çã♦ ❛♣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♦✿ ❆♥á❧✐s❡✳ ❆♣r♦✈❛❞❛ ♣♦r✿
Pr♦❢✳❉r ◆❛♣♦❧❡ó♥ ❈❛r♦ ❚✉❡st❛ ✲ ❯❋P❇ ✭❖r✐❡♥t❛❞♦r✮
Pr♦❢✳❉r ❆♥t♦♥✐♦ ❞❡ ❆♥❞r❛❞❡ ❡ ❙✐❧✈❛ ✲ ❯❋P❇
Pr♦❢✳❉r ❚✉rí❜✐♦ ❏♦sé ●♦♠❡s ❞♦s ❙❛♥t♦s ✲ ❯❋P❇
❆❣r❛❞❡❝✐♠❡♥t♦s
❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ♣r♦❢❡ss♦r ◆❛♣♦❧é♦♥ ❈❛r♦ ❚✉❡st❛✱ ♣❡❧❛s ♦♣♦rt✉♥❛s s✉❣❡stõ❡s ❡ ❧✉③ q✉❛♥❞♦ ❢✉✐ ❛❝♦♠❡t✐❞♦ ♣❡❧❛ ❡s❝✉r✐❞ã♦ ❞❛ ❞ú✈✐❞❛✱ q✉❡ ❢♦r❛♠ s✉st❡♥tá❝✉❧♦s ♣❛r❛ ❝❛♠✐♥❤❛❞❛ ✜r♠❡ ❞❛ ♠✐♥❤❛ ✐♥✈❡st✐❣❛çã♦✳
❆♦s ♣r♦❢❡ss♦r❡s✱ ❆♥tô♥✐♦ ❞❡ ❆♥❞r❛❞❡ ❡ ❙✐❧✈❛ ❡ ❚✉rí❜✐♦ ❙❛♥t♦s✱ ♣❡❧♦ ❛♣♦✐♦ ❡ ❝♦♥s❡❧❤♦s ❞❛❞♦s ♣❛r❛ r❡❛❧✐③❛çã♦ ❞❡ss❡ tr❛❜❛❧❤♦✳
➚ ♠✐♥❤❛ ❡s♣♦s❛ ❑é③✐❛ ❖❧✐✈❡✐r❛ ❈❛❜r❛❧ ✱ ❛ ♠✐♥❤❛ ✜❧❤❛ ▲❡tí❝✐❛ ▼♦♥r♦❡ ❈❛❜r❛❧✱ ❡ ♠✐♥❤❛ ♠ã❡ ▼❛r✐❛ ❏♦sé ❝r✐❛t✉r❛s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❞❛ ♠✐♥❤❛ ✈✐❞❛✱ s❡♠ ❛s q✉❛✐s ❡✉ ♥ã♦ s❡r✐❛ ♥❛❞❛✳
❉❡❞✐❝❛tór✐❛
❉❡❞✐❝♦ ❡st❡ tr❛❜❛❧❤♦ ❛♦s ♣r♦❢❡ss♦r❡s ❡ ❛♠✐❣♦s q✉❡✱ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡ ♠❡ ❛❥✉❞❛r❛♠ ❛ ❝♦♥❝r❡t✐③á✲❧♦✳
✏❆ ♠❛t❡♠át✐❝❛✱ ✈✐st❛ ❝♦rr❡t❛♠❡♥t❡✱ ♣♦ss✉✐ ♥ã♦ ❛♣❡♥❛s ✈❡r❞❛❞❡✱ ♠❛s t❛♠❜é♠ s✉♣r❡♠❛ ❜❡❧❡③❛ ✲ ✉♠❛ ❜❡❧❡③❛ ❢r✐❛ ❡ ❛✉st❡r❛✱ ❝♦♠♦ ❛ ❞❛ ❡s❝✉❧t✉r❛✳✑
❇❡rtr❛♥❞ ❘✉ss❡❧❧
❘❡s✉♠♦
❊st❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♠♦str❛r ❛ ❊①✐stê♥❝✐❛ ❡ ❛ ❯♥✐❝✐❞❛❞❡ ❞♦ ❈♦r♣♦ ❞♦s ◆ú♠❡r♦s ❘❡❛✐s✱ ✉s❛♥❞♦ ♣❛r❛ ✐ss♦✱ ♦s ❈♦rt❡s ❞❡ ❉❡❞❡❦✐♥❞ ❡ ♦ t❡♦r❡♠❛ ❞❛ ❞❡✜✲ ♥✐çã♦ ♣♦r ❘❡❝✉rsã♦✳ P❛r❛ ❝✉♠♣r✐r♠♦s t❛❧ ♦❜❥❡t✐✈♦✱ ❞❡✜♥✐♠♦s ❛ ♥♦çã♦ ❞❡ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞ ❡ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♠❛s ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s❀ ❡♠ s❡❣✉✐❞❛✱ ❛♣r❡s❡♥t❛♠♦s ❛s ♥♦çõ❡s ❞❡ ❈♦r♣♦✱ ❈♦r♣♦ ❖r❞❡♥❛❞♦ ❡ ❆rq✉✐♠❡❞✐❛♥♦✱ ❈♦r♣♦ ❖r❞❡♥❛❞♦ ❈♦♠♣❧❡t♦ ❡✱ ✜♥❛❧♠❡♥t❡✱ ❡♥✉♥❝✐❛♠♦s ❡ ❞❡♠♦♥str❛♠♦s ♦ ❚❡♦r❡♠❛ ❞❛ ❯♥✐❝✐❞❛❞❡ ❞♦ ❈♦r♣♦ ❞♦s ◆ú♠❡r♦s ❘❡❛✐s✳ ✳
P❛❧❛✈r❛s✲❈❤❛✈❡s✿
◆ú♠❡r♦s ❘❡❛✐s❀ ❈♦rt❡s ❞❡ ❉❡❞❡❦✐♥❞❀ ❈♦r♣♦ ❖r❞❡♥❛❞♦ ❈♦♠♣❧❡t♦❀ ■s♦♠♦r✜s♠♦❀
❆❜str❛❝t
❚❤✐s ✇♦r❦ ❛✐♠s t♦ s❤♦✇ t❤❡ ❡①✐st❡♥❝❡ ❛♥❞ ❯♥✐q✉❡♥❡ss ♦❢ t❤❡ ✜❡❧❞ ♦❢ ❘❡❛❧ ◆✉♠✲ ❜❡rs✱ ✉s✐♥❣ ❢♦r t❤✐s✱ ❉❡❞❡❦✐♥❞✬ ❈✉ts t❤❡♦r❡♠ ❛♥❞ t❤❡ ❉❡✜♥✐t✐♦♥ ❜② ❘❡❝✉rs✐♦♥✳❚♦ ❢✉❧✜❧❧ ❤✐s ❣♦❛❧✱ ✇❡ ❞❡✜♥❡ t❤❡ ♥♦t✐♦♥ ♦❢ ❉❡❞❡❦✐♥❞ ❈✉t ❛♥❞ ♣r❡s❡♥t s♦♠❡ ♦❢ ✐ts ♣r♦♣❡r✲ t✐❡s❀ t❤❡♥ ✐♥tr♦❞✉❝❡ t❤❡ ♥♦t✐♦♥s ♦❢ ❆r❝❤✐♠❡❞❡❛♥ ❖r❞❡r❡❞ ❛♥❞ ❋✐❡❧❞✱ ❈♦♠♣❧❡t❡ ❋✐❡❧❞ ❙♦rt❡❞ ❛♥❞ ✜♥❛❧❧② ❛rt✐❝✉❧❛t❡ ❛♥❞ ❞❡♠♦♥str❛t❡ t❤❡ ❯♥✐q✉❡♥❡ss ❚❤❡♦r❡♠ ♦❢ ❋✐❡❧❞ ❘❡❛❧ ◆✉♠❜❡rs✳
✳
❑❡② ❲♦r❞s✿
❘❡❛❧ ◆✉♠❜❡rs✱ ❉❡❞❡❦✐♥❞ ❈✉t✱ ❈♦♠♣❧❡t❡ ❋✐❡❧❞ ❙♦rt❡❞✱■s♦♠♦r♣❤✐s♠✳
❙✉♠ár✐♦
✶ ❈♦rt❡s ❞❡ ❉❡❞❡❦✐♥❞ ✶
✷ ❈♦♥str✉çã♦ ❞♦s ◆ú♠❡r♦s ❘❡❛✐s ✶✵
✸ Pr♦♣r✐❡❞❛❞❡s ❆❧❣é❜r✐❝❛s ❞♦s ◆ú♠❡r♦s r❡❛✐s ✷✷
✹ ❈♦r♣♦ ❖r❞❡♥❛❞♦ ✷✾
✹✳✶ ■♥t❡r✈❛❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✺ ❈♦r♣♦ ❖r❞❡♥❛❞♦ ❈♦♠♣❧❡t♦ ✸✽
✻ ❯♥✐❝✐❞❛❞❡ ❞♦s ◆ú♠❡r♦s ❘❡❛✐s ✹✾
✻✳✶ ❉❡✜♥✐çã♦ ♣♦r ❘❡❝♦rrê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵
✼ ❈♦♥s✐❞❡r❛çõ❡s ❤✐stór✐❝❛s ✻✵
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✻✻
■♥tr♦❞✉çã♦
❖ ❝♦♥❝❡✐t♦ ❞❡ ♥ú♠❡r♦ r❡❛❧ ❡stá ❛ss♦❝✐❛❞♦ ❞❡s❞❡ ❛ ✐❞❡✐❛ ❞❡ ❝♦♥t❛❣❡♠ ❞❡ ♦❜❥❡t♦s✱ ❝♦✐s❛s ❡ ❡t❝ ✭♥♦çã♦ ❞❡ q✉❛♥t✐❞❛❞❡✮ à ❞❡ ❧♦❝❛❧✐③❛çã♦ ❞❡ ♣♦♥t♦s ❞❛ r❡t❛ ✭♥♦çã♦ ❣❡♦✲ ♠étr✐❝❛✮ ♦✉ ❛té ♠❡s♠♦ ❞❡ ❡♥t❡s q✉❡ s❛t✐s❢❛③❡♠ ✉♠ ❝♦r♣♦ ❞❡ ♣r♦♣r✐❡❞❛❞❡s ✭♥♦çã♦ ❛❜str❛t❛✮✳ ❊♥q✉❛♥t♦ q✉❡ ❛ ♥♦çã♦ q✉❛♥t✐t❛t✐✈❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s ❡ ♠❡s♠♦ ❛ ❣❡♦♠é✲ tr✐❝❛ r❡s♦❧✈❡ ♠✉✐t♦ ❜❡♠ ♦s ♣r♦❜❧❡♠❛s ❡ ❞❡♠❛♥❞❛s ❞♦ ♠✉♥❞♦ r❡❛❧❀ ❛ ♥♦çã♦ ❛❜str❛t❛ r❡s♦❧✈❡ ❝♦♠ ♠✉✐t❛ ♣r❡❝✐sã♦ às q✉❡stõ❡s ❢✉♥❞❛♠❡♥t❛✐s ❞❛ ♣ró♣r✐❛ ♠❛t❡♠át✐❝❛✱ ❡s✲ t❛❜❡❧❡❝❡♥❞♦ ✉♠❛ ✈✐sã♦ ♠❛✐s ♣r♦❢✉♥❞❛ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ♥ú♠❡r♦✳ ◆♦ ✜♥❛❧ ❞♦ sé❝✉❧♦ ■❳❳✱ ❘✐❝❤❛r❞ ❉❡❞❡❦✐♥❞ s❡♥t✐✉ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ✉♠❛ ✐♥✈❡st✐❣❛çã♦ ♠❛✐s r✐❣♦r♦s❛ ❞♦s ♥ú♠❡r♦s r❡❛✐s ♣❛r❛ ❥✉st✐✜❝❛r ❝❡rt♦s r❡s✉❧t❛❞♦s ❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧ ❞❡s✲ ❝♦❜❡rt♦s ❞❡s❞❡ à s✉❛ ❝r✐❛çã♦ ♣♦r ■s❛❛❝ ◆❡✇t♦♥ ❡ ▲❡✐❜♥✐③✳❆ss✐♠✱ t❡✈❡✲s❡ ❛ ✐❞❡✐❛ ❞❡ ❢✉♥❞❛♠❡♥t❛r ♦ ❝♦♥❝❡✐t♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s✱ ❣r❛ç❛s ❛♦s tr❛❜❛❧❤♦s ❞❡s❡♥✈♦❧✈✐❞♦s ♣♦r ❘✐❝❤❛r❞ ❉❡❞❡❦✐♥❞ ✭✶✽✶✸✲✶✾✶✻✮✱ ●❡♦r❣ ❈❛♥t♦r ✭✶✽✹✺✲✶✾✶✽✮ ❡ ●✐✉s❡♣♣❡ P❡❛♥♦ ✭✶✽✺✽✲ ✶✾✸✷✮✳ ❆ ♣❛rt✐r ❞♦s tr❛❜❛❧❤♦s ❞❡ss❡s ♠❛t❡♠át✐❝♦s✱ ❛ ♥♦çã♦ ❞❡ ♥ú♠❡r♦ r❡❛❧ s❡ t♦r♥♦✉ ♠❛✐s ♣r❡❝✐s❛✳
❊♠ ❣❡r❛❧✱ ❛ ✐❞❡✐❛ ❜ás✐❝❛ ❞❛ ❈♦♥str✉çã♦ ❞❡ ◆ú♠❡r♦s ❘❡❛✐s ❝♦♥s✐st❡ ❡♠ ♣❛rt✐r ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ♣r❡✈✐❛♠❡♥t❡ ❡st❛❜❡❧❡❝✐❞♦✱ ♣♦r ✉♠ ❝♦r♣♦ ❞❡ ❛①✐♦♠❛s✱❡ ❡♠ s❡❣✉✐❞❛✱ ❛✈❡r✐❣✉❛r ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ s✉❛s ♦♣❡r❛çõ❡s q✉❡ ❞❡❧❡ s✉s❝✐t❛♠ ✭♦♣❡r❛çõ❡s ❡ss❛s ❡st❛✲ ❜❡❧❡❝✐❞❛s ❧♦❣♦ ❞❡♣♦✐s ❞♦s ❛①✐♦♠❛s✮❀ s❡ ♣♦r ❛❧❣✉♠ ♠♦t✐✈♦✱ s✉r❣✐r ✉♠❛ ♥♦✈❛ ♦♣❡r❛çã♦ ✉♥ár✐❛ ♦✉ ❜✐♥ár✐❛ ❡♠ q✉❡ ♥ã♦ é ♣♦ssí✈❡❧ ❞❡ s❡r ❡①❡❝✉t❛❞❛ ♣♦r ♣❡❧♦ ♠❡♥♦s ✉♠ ❡❧❡♠❡♥t♦ ♦✉ ♣❛r ❞❡ ❡❧❡♠❡♥t♦s✱ ❡♥tã♦ ❢❛③✲s❡ ♥❡❝❡ssár✐♦ ❝♦♥t♦r♥❛r ❡ss❡ ♣r♦❜❧❡♠❛ ❝r✐❛♥❞♦✲s❡ ✉♠ ♥♦✈♦ ❡❧❡♠❡♥t♦✱❡ ♣♦rt❛♥t♦✱ ✉♠ ♥♦✈♦ ❝♦♥❥✉♥t♦✱ ❛ ♣❛rt✐r ❞❡ss❛ ♥♦✈❛ ♦♣❡r❛çã♦✳ ❆ss✐♠✱ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞♦ ❛♥t✐❣♦ ❡ ❞♦ ♥♦✈♦ ❝♦♥❥✉♥t♦ ♣❛ss❛♠ ❛ s❡r ❡s❝r✐t♦s ❡♠ t❡r♠♦s ❞❡ss❛ ♥♦✈❛ ♦♣❡r❛çã♦✳ ❖s ❡❧❡♠❡♥t♦s ❞❡ss❡ ♥♦✈♦ ❝♦♥❥✉♥t♦ ♣♦ss✉❡♠ ♥❛t✉r❡③❛ ❞✐✈❡rs❛ ❛♦s ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦ ❛♥t❡r✐♦r✱ ♣♦ré♠ ♣♦ss✉❡♠ ❛s ♠❡s♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡❧❡✱ ❛❧é♠ ❞❛s ♥♦✈❛s ♦♣❡r❛çõ❡s ❡ ♣r♦♣r✐❡❞❛❞❡s q✉❡ ❞❡❧❡ ♣♦ss❛♠ s✉r❣✐r✿ ❡ss❡ é ♦ ✈❡r❞❛❞❡✐r♦ s❡♥t✐❞♦ ❞❛ ❈♦♥str✉çã♦ ❞❡ ◆ú♠❡r♦s✳
❆♦ s❡ ❝♦♥str✉✐r ◆ú♠❡r♦s ❘❡❛✐s✱ ♣♦❞❡♠♦s ✉s❛r ❞♦✐s ♠♦❞♦s ❞✐st✐♥t♦s ❞❡ ❛①✐♦♠❛✲ t✐③❛çã♦✿ ♦ ♣r✐♠❡✐r♦ ♠♦❞♦ é ❛ tr✐❧♦❣✐❛ N−Z−Q✱ ♠❛✐s ❧♦♥❣♦ ❡ ❝♦♠♣❧❡t♦✱ ❝♦♥s✐st❡ ❡♠ ❝♦♥str✉✐r ♦ ❝♦♥❥✉♥t♦ ❞♦s ◆ú♠❡r♦s ◆❛t✉r❛✐s ♣❛rt✐♥❞♦ ❞♦s ❛①✐♦♠❛s ❞❡ P❡❛♥♦✱ ❞❡✲ ✜♥✐♥❞♦ ❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ❡st❛❜❡❧❡❝❡♥❞♦ ❛ r❡❧❛çã♦ ❞❡ ♦r❞❡♠✱ ❞❡❞✉③✐♥❞♦✱ ❛ ♣❛rt✐r ❞✐ss♦✱ ✈ár✐❛s ♣r♦♣r✐❡❞❛❞❡s r❡❧❛t✐✈❛s ❛ ❡❧❛s✱ s❡❣✉✐♥❞♦✱ ❧♦❣♦ ❞❡✲ ♣♦✐s✱ à ❝♦♥str✉çã♦ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ◆ú♠❡r♦s ■♥t❡✐r♦s✱ ✉s❛♥❞♦ ❛ ♥♦çã♦ ❞❡ ❈❧❛ss❡s ❞❡
❊q✉✐✈❛❧ê♥❝✐❛ ❞❡✜♥✐♥❞♦ s♦❜r❡ ❡❧❛s ❛s ♦♣❡r❛çõ❡s ❜✐♥ár✐❛s ❞❡ ❛❞✐çã♦✱ ♠✉❧t✐♣❧✐❝❛çã♦❀ ❛ ♦♣❡r❛çã♦ ✉♥ár✐❛ ❞❡ ❡❧❡♠❡♥t♦ ♦♣♦st♦ ❡ ❡st❛❜❡❧❡❝❡♥❞♦ ❛ r❡❧❛çã♦ ❞❡ ♦r❞❡♠ ❡♥tr❡ s❡✉s ❡❧❡♠❡♥t♦s ❡ ❞❡❞✉③✐♥❞♦ s✉❛s ❝♦♥s❡q✉ê♥❝✐❛s❀ ❝❛♠✐♥❤♦ ❛①✐♦♠át✐❝♦ s❡♠❡❧❤❛♥t❡ é ✉s❛❞♦✱ t❛♠❜é♠✱ ♣❛r❛ ❝♦♥str✉✐r ♦ ❝♦♥❥✉♥t♦ ❞♦s ◆ú♠❡r♦s ❘❛❝✐♦♥❛✐s❀ ♦ s❡❣✉♥❞♦✱ é ❛ tr✐❧♦❣✐❛ Z−N−Q✱ r❡❧❛t✐✈❛♠❡♥t❡ ❛♦ ❛♥t❡r✐♦r é ♠❛✐s ❝✉rt♦✱ ♥❡❧❡ ❛❞♠✐t❡✲s❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ❝♦♠♦ ❛①✐♦♠❛s ❝❛r❛❝t❡r✐③❛♥❞♦✲♦ ❝♦♠♦ ✉♠ ❞♦♠í♥✐♦ ❞❡ ■♥t❡❣r✐✲ ❞❛❞❡ ❖r❞❡♥❛❞♦✱ ❡♠ s❡❣✉✐❞❛✱ sã♦ ♣r♦✈❛❞♦s ❛s ♣r✐♥❝✐♣❛✐s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ◆ú♠❡r♦s ■♥t❡✐r♦s ❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s ❛❧❣✉♠❛s ❞❛s q✉❛✐s ❝♦♥s✐❞❡r❛❞❛s✱ ❛♥t❡r✐♦r♠❡♥t❡✱ ❝♦♠♦ ❛①✐♦♠❛s ❞❡ P❡❛♥♦❀ ❝♦♥t✉❞♦✱ ♣♦ré♠ ❛ ❝♦♥str✉çã♦ ❞♦s ◆ú♠❡r♦s ❘❛❝✐♦♥❛✐s ❝♦♥t✐♥✉❛ s❡♥❞♦ ❛ ♠❡s♠❛ q✉❡ ❛ ❛♥t❡r✐♦r✳
❯♠❛ ✈❡③ s❡❣✉✐❞♦ ✉♠ ❞♦s ❝❛♠✐♥❤♦s ❞❛ tr✐❧♦❣✐❛✱ ♦ ♣❛ss♦ s❡❣✉✐♥t❡ ❝♦♥s✐st❡ ✜♥❛❧✐✲ ③❛r ❛ ❝♦♥str✉çã♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ♣❛rt✐♥❞♦ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ◆ú♠❡r♦s ❘❛❝✐♦♥❛✐s✳ ❊①✐st❡ ❞♦✐s ❝❛♠✐♥❤♦s tr❛❞✐❝✐♦♥❛✐s✿ ✉♠ ❞❡❧❡s ❝♦♥s✐st❡ ❡♠ ✉s❛r ♦s ❝♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞ ✲ ❡ss❡ ♠ét♦❞♦ ❢♦✐ ❝r✐❛❞♦ ♣❡❧♦ ♠❛t❡♠át✐❝♦ q✉❡ ❧❡✈❛ s❡✉ ♥♦♠❡❀ ♦ ♦✉tr♦✱ ❝♦♥s✐st❡ ❡♠ ✉s❛r ❛s s❡q✉ê♥❝✐❛s ❞❡ ❈❛✉❝❤② ✲ ♠ét♦❞♦ ❝r✐❛❞♦ ♣❡❧♦ ♠❛t❡♠át✐❝♦ ●❡♦r❣❡ ❈❛♥t♦r✳ ❱❛❧❡ à ♣❡♥❛ ❢r✐s❛r✱ q✉❡ ❛♠❜♦s ♦s ♠ét♦❞♦s ♣❛rt❡♠ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ◆ú♠❡r♦s ❘❛❝✐♦♥❛✐s✳ ❆❧é♠ ❞❡❧❡s✱ ❡①✐st❡♠ ♦✉tr♦s✱ ❛ s❛❜❡r✿ ✉s❛♥❞♦ à ♥♦çã♦ ❞❡ ◗✉❛♥t✐❞❛❞❡✱ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❈❧❛ss❡s ❞❡ ❉❡❝❧✐✈❡s ❡ ❛ ♥♦çã♦ ❞❡ ❙✉❝❡ssã♦ ❞❡ ■♥t❡r✈❛❧♦s ❊♥❝❛✐①❛❞♦s ❡✱ ❛✐♥❞❛✱ ❛ ✐❞❡✐❛ ❞❡ s✉❝❡ssã♦ ❞❡ ◆ú♠❡r♦s ❘❛❝✐♦♥❛✐s ❉❡❝✐♠❛✐s✳ ❉❡♥tr❡ ❡ss❛s ❝♦♥str✉çõ❡s✱ ♦♣t❛♠♦s ♣♦r ❞❡s❝r❡✈❡r ❛ ❝♦♥str✉çã♦ ✈✐❛ ❈♦rt❡s ❞❡ ❉❡❞❡❦✐♥❞✱ ✐ss♦ é ❞❡✈✐❞♦ ❛ ✉♠ ♠♦t✐✈♦ ❜ás✐❝♦✿ é q✉❡ ❡ss❡ ♠ét♦❞♦ ❞❡♣❡♥❞❡ ❛♣❡♥❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♥❛t✉r❛❧ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ◆ú✲ ♠❡r♦s ❘❛❝✐♦♥❛✐s ❞✐s♣❡♥s❛♥❞♦ ❛s ❞❡✜♥✐çõ❡s ❡ ♣r♦♣r✐❡❞❛❞❡s ❞❡♠❛s✐❛❞❛♠❡♥t❡ ❧♦♥❣❛s ❞❡ s❡q✉ê♥❝✐❛s ♥✉♠ér✐❝❛s ✈✐st❛s✱ ♣♦r ❡①❡♠♣❧♦✱ ♥♦ ♠ét♦❞♦ ❞❡ ❝♦♥str✉çã♦ ♣♦r s❡q✉ê♥❝✐❛s ❞❡ ❈❛✉❝❤②❀ ✐ss♦ t♦r♥❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ♠❛✐s ❞✐❞át✐❝♦ ❡ ❝✉rt♦✳ P♦r ✜♠✱ ❛❞♠✐t✐♥❞♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ❝♦♠♦ ✉♠ ❝♦r♣♦ ♦r❞❡♥❛❞♦ ❝♦♠♣❧❡t♦✱ ♣r♦✈❛r❡♠♦s s✉❛ ❯♥✐❝✐❞❛❞❡ ❡ ❝♦♠♣❧❡♠❡♥t❛r❡♠♦s ❛ ❝♦♥str✉çã♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s r❡♣r❡s❡♥t❛♥❞♦✲♦s ♣♦r ♠❡✐♦ ❞❛ ❡①♣❛♥sã♦ ❞❡❝✐♠❛❧✳ ◆❡st❡ tr❛❜❛❧❤♦ s♦❜r❡ ❝♦♥str✉çã♦ ✭❡ ♣♦rt❛♥t♦✱s♦❜r❡ ❡①✐s✲ tê♥❝✐❛✮ ❞♦s ♥ú♠❡r♦s r❡❛✐s ✉s❛♥❞♦ ❈♦rt❡s ❞❡ ❉❡❞❡❦✐♥❞ ❞✐s♣❡♥s❛r❡♠♦s ♦ ❝❛♠✐♥❤♦ q✉❡ ❧❡✈❛ ❝♦♠❡ç❛♥❞♦ ♣❡❧♦ ❝♦♥❥✉♥t♦ ❞♦s ◆ú♠❡r♦s ◆❛t✉r❛✐s ✉s❛♥❞♦ ❛ ❆①✐♦♠át✐❝❛ ❞❡ P❡✲ ❛♥♦✱ ❡ s✉❛ ♣♦st❡r✐♦r ❝♦♥str✉çã♦ ❞♦s ◆ú♠❡r♦s ■♥t❡✐r♦s ❡ ❘❛❝✐♦♥❛✐s✳ ❆❞♠✐t✐♠♦s q✉❡ ♦ ❧❡✐t♦r ❥á t❡♥❤❛ ✉♠❛ ♥♦çã♦ ❞❛ tr✐❧❤❛ ❛①✐♦♠át✐❝❛ ❞❛ tr✐❧♦❣✐❛ N−Z−Q♦✉Z−N−Q✳
❈❛♣ít✉❧♦ ✶
❈♦rt❡s ❞❡ ❉❡❞❡❦✐♥❞
❆♦ ❝♦♥str✉✐r♠♦s ♦s ♥ú♠❡r♦s r❡❛✐s ✉s❛♥❞♦ ♦s ❈♦rt❡s ❞❡ ❉❡❞❡❦✐♥❞✱ ♣❛rt✐r♠♦s ❞♦ ♣r❡ss✉♣♦st♦ ❞❡ q✉❡ ❝♦♥❤❡❝❡♠♦s ❡ ❛❝❡✐t❛♠♦s t♦❞❛s ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✱❤❡r❞❛❞❛s ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡✱ ❞♦s ❛①✐♦♠❛s ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❡ ✐♥t❡✐r♦s✳ ❆ ♥♦çã♦ q✉❡ ❛❞♦t❛r❡♠♦s ♣❛r❛ ♦ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞ ✱♥❡st❡ tr❛❜❛❧❤♦✱é ❛q✉❡❧❛ q✉❡ ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ ❞❡ ❇❡rtr❛♥❞ ❘✉ss❡❧✳
❉❡✜♥✐çã♦ ✶✳✶ ❙❡❥❛ ✉♠ s✉❜❝♦♥❥✉♥t♦ A⊂ Q✳ ❉✐③❡♠♦s q✉❡ A é ✉♠ ❈♦rt❡ ❞❡ ❉❡❞❡✲
❦✐♥❞ s❡ ♣♦ss✉✐ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿
■✳ A6=∅ ❡ A6=Q✱ ✐st♦ é✱ A é ✉♠ s✉❜❝♦♥❥✉♥t♦ ♥ã♦✲✈❛③✐♦ ❡ ♣ró♣r✐♦ ❞❡ Q❀
■■✳ ❉❛❞♦ q✉❡ x∈ A✱ s❡ y ∈Q ❡ q✉❡ y ≥x✱ ❡♥tã♦ y ∈A✱ ✐st♦ é✱ t♦❞♦ ❡❧❡♠❡♥t♦ y
❞❡ Q ♠❛✐♦r ❞♦ q✉❡ ♦✉ ❛ ✐❣✉❛❧ ✉♠ ❡❧❡♠❡♥t♦ x❞❡ A é✱ t❛♠❜é♠✱ ❡❧❡♠❡♥t♦ ❞❡ A❀
■■■✳ ❉❛❞♦ q✉❡x∈A✱ ❡①✐st❡ ✉♠y∈A t❛❧ q✉❡y < x✱ ✐st♦ é✱ ♣❛r❛ q✉❛❧q✉❡r ❡❧❡♠❡♥t♦ x ❞❡ A s❡♠♣r❡ ❡①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ ❞❡❧❡ y ♠❡♥♦r q✉❡ x✳
❖ ▲❡♠❛ s❡❣✉✐♥t❡ ♣r♦✈❛ ❛ ❡①✐stê♥❝✐❛ ❞♦ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞✳
▲❡♠❛ ✶✳✶ ❙❡❥❛ r∈Q✳ ❖ ❝♦♥❥✉♥t♦ D={x∈Q|x > r} é ✉♠ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞✳
❉❡♠♦♥str❛çã♦✿ Pr♦✈❛r❡♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ ❛❝✐♠❛ s❛t✐s❢❛③ ❛s três ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞✳
■✳ ❉❛❞♦ q✉❡ r ∈ Q s❡❣✉❡✲s❡ ✐♠❡❞✐❛t❛♠❡♥t❡ q✉❡ r−1, r+ 1 ∈ Q ❡ q✉❡ r−1< r < r+ 1✱ ❞♦ q✉❛❧ t❡♠♦s q✉❡ r+ 1∈Q❡ r−1∈/ Q✳ P♦rt❛♥t♦✱ t❡♠♦sQ6=∅❡
D6=Q❀
■■✳ ❉❛❞♦ q✉❡x∈D❡ s✉♣♦♥❞♦ ❞❛❞♦ ✉♠y∈Qt❛❧ q✉❡y > xt❡♠♦s q✉❡y > x > r✱
♣♦✐s✱ x∈D✳ P♦rt❛♥t♦✱ y∈D❀
■■■✳ ❉❛❞♦ q✉❡x∈Dt❡♠♦s q✉❡r < x✳ ❚♦♠❛♥❞♦ ✉♠y = (r+2x) t❡♠♦sr < (r+2x) < x
❡✱ ♣♦rt❛♥t♦✱ y∈D✳
❉❡✜♥✐çã♦ ✶✳✷ ❙❡❥❛ r ∈ Q✳ ❉❡✜♥✐♠♦s ✉♠ ❈♦rt❡ ❘❛❝✐♦♥❛❧ r❡❧❛t✐✈♦ ❛ r✱ ♦ q✉❛❧
❞❡♥♦t❛♠♦s ♣♦r Dr✱ ♦ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞
Dr ={r∈Q|x > r}.
❖❜s❡r✈❛çã♦ ✶✳✷✳✶ ❊♠❜♦r❛ t❡♥❤❛♠♦s ✉♠❛ ❞❡✜♥✐çã♦✱ ❛ ♣r✐♦r✐✱ ❞❡ ❈♦rt❡ ■rr❛❝✐♦♥❛❧ ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ♠❛✐♦r❡s ❞♦ q✉❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ q✉❡ ♥ã♦ é r❛❝✐♦♥❛❧✱ ♦ ✉s♦ ❞❡❧❛ ♥❡st❡ ♠♦♠❡♥t♦ ♥ã♦ ♣❡r♠✐t❡ ✉♠❛ ♣r♦✈❛ q✉❡ ❡❧❡ s❡❥❛ ✉♠ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞✱ ♣♦✐s✱ ❛✐♥❞❛ ♥ã♦ ❞❡✜♥✐♠♦s ♦ q✉❡ s✐❣♥✐✜❝❛ ♥ú♠❡r♦ r❡❛❧✳ P♦r ❡①❡♠♣❧♦✱ ♦ ❝♦♥❥✉♥t♦ D = {x ∈ Q|x > √2} ❢❛③ ♦ ✉s♦ ❞❡ ❛❧❣♦ q✉❡ ❛✐♥❞❛ ♥ã♦ ❢♦✐ ❞❡✜♥✐❞♦✱ ♦ ♥ú♠❡r♦ √2 ✳ ❚❡rí❛♠♦s q✉❡ ♣r♦✈❛r✱ ♣r✐♠❡✐r♦✱ q✉❡ ❡ss❡ ♥ú♠❡r♦ r❡❛❧ ❡①✐st❡ ❡✱ ❛✐♥❞❛
♠❛✐s✱ ♣r♦✈❛r q✉❡ ❡❧❡ ♥ã♦ é r❛❝✐♦♥❛❧✳ P♦❞❡♠♦s s✐♠✱ ❞❡s❝r❡✈❡r ♦ ❝♦♥❥✉♥t♦ ❞❛❞♦ ♥❛ ❞❡✜♥✐çã♦ ❞❡ ✉♠❛ ❢♦r♠❛ ❡q✉✐✈❛❧❡♥t❡ ❢❛③❡♥❞♦ ♦ ✉s♦ ❞❡ ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ♦ ❝♦♥❥✉♥t♦ D = {x ∈ Q|x > 0 ❡ x2 > 2}✿ t❛❧ ❝♦♥❥✉♥t♦ é ❡q✉✐✈❛❧❡♥t❡ ❛ D ={x∈ Q|x >√2}✳ ❆té ❛q✉✐ ❢❛r❡♠♦s ❛♣❡♥❛s✱ ❡ ♥❛❞❛ ♠❛✐s✱ ❛ ♣r♦✈❛ ❞❡ q✉❡ ❡ss❡
❝♦♥❥✉♥t♦ ♥❛ s✉❛ ❢♦r♠❛ ♠❛✐s ❣❡r❛❧✱ D={x∈Q|x >0
❡
xn > p
} ♦♥❞❡ p ∈ Q✱ é ✉♠ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞✱ ♥❛ ♣r♦♣♦s✐çã♦ ❛❜❛✐①♦✳ ❆♥t❡s ❞❡
❞❡♠♦♥strá✲❧♦✱ ❡♥✉♥❝✐❛r❡♠♦s✱ s❡♠ ❞❡♠♦♥str❛çã♦✱ três ❧❡♠❛s út❡✐s q✉❡ ❧❡✈❛rã♦ ❛ ❞❡✲ ♠♦♥str❛çã♦ ❞❡st❛ ❞❡st❛ ♣r♦♣r✐❡❞❛❞❡ ❡ ❞❡ ♦✉tr❛ ♥ã♦ ♠❡♥♦s ✐♠♣♦rt❛♥t❡✳
▲❡♠❛ ✶✳✷ ❉❛❞♦ x ∈ Q ❝♦♠ x > −1 ❡ n ∈ N✳ ❊♥tã♦ ✈❛❧❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❇❡r♥♦✉❧❧✐ (1 +x)n≥1 +nx✳
▲❡♠❛ ✶✳✸ ❙❡❥❛ ✉♠ r∈ Q✱ r >0✱ ❡ ✵❁❞❁✶✳ ❊♥tã♦ ♣❛r❛ ❝❛❞❛ r ❡ n∈ N ❡①✐st❡ ✉♠
An✱ ❞❡♣❡♥❞❡♥❞♦ ❞❡ r✱ t❛❧ q✉❡ (r+d)n ≤rn+A nd✳
▲❡♠❛ ✶✳✹ ❙❡❥❛ r >0✱ ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧✳ ❙❡ rn < p✱ ❡♥tã♦ ❡①✐st❡ ✉♠ k
∈N t❛❧
q✉❡ s❡ t❡♠✱ ❛✐♥❞❛✱ (r+1k)n< p✳
Pr♦♣♦s✐çã♦ ✶✳✶ ❖ ❝♦♥❥✉♥t♦ M ⊆Q ❞❡✜♥✐❞♦ ♣♦r M ={x ∈Q|x >0 ❡ xn > p}✱ ♦♥❞❡ p∈Q✱ p >0✱ é ✉♠ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞✳
❉❡♠♦♥str❛çã♦✿ ❚❡♠♦s ❞♦✐s ❝❛s♦s ♣♦ssí✈❡✐s✿ ✭❛✮ ✵❁♣❁✶❀
✭❜✮ ♣❃✶❀
❊ss❡s ❞♦✐s ❝❛s♦s ♣♦ssí✈❡✐s s❡r✈✐rã♦ ❛♣❡♥❛s ♣❛r❛ ❞❡♠♦♥str❛r♠♦s ❛ ♣r✐♠❡✐r❛ ♣❛rt❡ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❈♦rt❡✳ ❱❡❥❛♠♦s✿
■✳ P❛rt❡ ✭❛✮✿ ❙❡0< p <1✱ ❡♥tã♦ 2−p∈M✱ ♣♦✐s✱ 2−p >1> p >0 ❡ q✉❡ (2−p)n>1> p✳ ▲♦❣♦✱ M 6=∅✳
P❛rt❡ ✭❜✮✿ s❡ p > 1 t❡♠♦s p+ 1 > p > 1 ♦ q✉❡ ✐♠♣❧✐❝❛ p+ 1 > 0 ❡ (p+ 1)n> r✳ P♦rt❛♥t♦✱ p+ 1
∈M✱ ✐st♦ é ✱ M 6=∅✳ ❈♦♠♦0∈/ M✱ t❡♠♦s M 6=Q✳
■■✳ ❆❣♦r❛✱ ♣r♦✈❛r❡♠♦s ❛ s❡❣✉♥❞❛ ♣❛rt❡ ❞❛ ❞❡✜♥✐çã♦✿ s✉♣♦♥❞♦ q✉❡ x ∈ M ❡ q✉❡ y≥x s❡❣✉❡✲s❡ q✉❡ y≥x >0 ✐♠♣❧✐❝❛yn
≥xn> p ❞❛ q✉❛❧ s❡ ❝♦♥❝❧✉✐ y
∈M✳
■■■✳ ❆❣♦r❛✱ ✈❛♠♦s ♣r♦✈❛r q✉❡ ❡①✐st❡ ✉♠ y ∈ M t❛❧ q✉❡ y < x✳ ❚❡♠♦s ❞♦✐s ❝❛s♦s
♣♦ssí✈❡✐s✿ ✭❛✮ x < xn−p
nxn−1✳ ✭❜✮ xn−p
nxn−1 < x✳
✭❛✮ ❙❡ t♦♠❛r♠♦s ✉♠y = x
2 t❡r❡♠♦s ♣❡❧❛ ❤✐♣ót❡s❡ ②❁
xn
−p
nxn−1✳ ❆ss✐♠✱ ✉s❛♥❞♦ ♦ ▲❡♠❛ ✶✳✷✱ t❡♠♦sp < xn−nyxn−1 =xn(1−ny
x)≤x
n 1− y x
n
= (x−y)n= x
2
n
< xn♦ q✉❡ ✐♠♣❧✐❝❛yn< xn✱ ❧♦❣♦✱y < x✳ P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛♥t❡r✐♦r ✱ t❡♠♦s yn > p✱ ❡ ♣❡❧♦ q✉❡ ❞❡✜♥✐♠♦s✱ y > 0✳ P♦rt❛♥t♦✱ ♣r♦✈❛♠♦s q✉❡ ❡①✐st❡ ✉♠y ∈M t❛❧ q✉❡ y < x✳
✭❜✮ P❡❧❛ ❤✐♣ót❡s❡✱ t♦♠❛♠♦s ✉♠ s t❛❧ q✉❡ 0 < s < xn−p
nxn−1 < x ❡ 0 < s < x n✳ ❋❛③❡♥❞♦ y = x −s✱ ✈❡♠♦s q✉❡ y > 0✳ ❙❡❣✉❡✲s❡ s✉❝❡ss✐✈❛♠❡♥t❡✱ q✉❡ p < xn−nsxn−1 = xn(1−ns
x) ≤ x
n(1− s x)
n = (x−s)n < xn ♦ q✉❡ ✐♠♣❧✐❝❛ y < x ❡ p < yn✳ P♦rt❛♥t♦✱ ♣r♦✈❛♠♦s q✉❡ ❡①✐st❡ ✉♠ y ∈ M t❛❧ q✉❡y < x✳
Pr♦♣♦s✐çã♦ ✶✳✷ ❙❡ M =Dr ♣❛r❛ ❛❧❣✉♠ r∈Q ❡♥tã♦ rn=p✳
❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ ♦ ❝♦♥trár✐♦✳❙❡❥❛ rn < p✳ ❙❡ r >0 ❡①✐st❡ ✉♠ k ∈N t❛❧ q✉❡ r+ 1kn
< p✱ ▲❡♠❛ ✶✳✹✳ ❈❤❛♠❛♥❞♦ y = r+ 1k t❡♠♦s y > r ❡ yn < p✳❙❡❣✉❡✲s❡ q✉❡✱ ❡①✐st❡ ✉♠ y ∈ Dr t❛❧ q✉❡ y /∈ M✳ P♦rt❛♥t♦✱ t❡♠♦s M 6= Dr✳ ❆❣♦r❛✱ s❡ r < 0 ❞❡✈❡♠♦s ❧❡✈❛r ❡♠ ❝♦♥s✐❞❡r❛çã♦ ❞♦✐s ❝❛s♦s ♣♦ssí✈❡✐s ♣❛r❛ n✿
✭❛✮ n é ♥❛t✉r❛❧ ♣❛r✳
✭❜✮ n é ♥❛t✉r❛❧ í♠♣❛r✳
✭❛✮ ◆♦t❡ q✉❡ (−r)n = rn < p✱ s❡ n é ♣❛r✳❈♦♠♦ −r > 0✱♣❡❧♦ ▲❡♠❛✶✳✹✱ ❡①✐st❡ ✉♠ k ∈ N t❛❧ q✉❡ −r+1kn
< p✳ ❈❤❛♠❛♥❞♦ y = −r+ 1k t❡♠♦s y > r ❡ yn < p✳ ❆ss✐♠✱ t❡♠♦s ❛ ✐♠♣❧✐❝❛çã♦ ❞❡ q✉❡ ❡①✐st❡ ✉♠y ∈Dr t❛❧ q✉❡y /∈M✳ P♦rt❛♥t♦✱ t❡♠♦s
M 6=D✳
✭❜✮◆♦t❡ q✉❡✱ ♥❡st❡ ❝❛s♦✱ rn < p ♣❛r❛ t♦❞♦ r < 0 ❡ n í♠♣❛r✳ ❈❤❛♠❛♥❞♦ y = r
2
t❡♠♦s y > r ❡ yn < p✱ ♣♦✐s✱ y < 0✳ P♦rt❛♥t♦✱ t❡♠♦s q✉❡ y ∈ D
r ✐♠♣❧✐❝❛ y /∈ M✳ ❆ss✐♠✱ ❡♠ ❛♠❜♦s ♦s ❝❛s♦s ✈❡r✐✜❝❛♠♦s q✉❡ M 6=Dr ♣❛r❛ t♦❞♦ r∈Q✳
❆❣♦r❛✱ s✉♣♦♥❤❛ rn> r✳ ❚❡♠♦s ❞♦✐s ❝❛s♦s✿
✭❛✮ ❙❡ r > 0 t❡♠♦s q✉❡ ❡①✐st❡ ✉♠ k ∈ N t❛❧ q✉❡ r− 1
k
n
> p✳ ❈❛s♦ t❡♥❤❛♠♦s r−1
k >0♣♦❞❡♠♦s t♦♠❛ry=r−
1
k ✳ ❆ss✐♠✱ t❡♠♦sy >0❡y
n> p✱ ❝♦♠y < r✳ P♦rt❛♥t♦✱ ♣r♦✈❛♠♦s q✉❡ ❡①✐st❡ y ∈ M t❛❧ q✉❡ y /∈ Dr✳ ❆❣♦r❛✱ s❡ t✐✈❡r♠♦s
y=r−1
k <0✱ t♦♠❛♠♦s ✉♠ y=r−
1
s✱ ❝♦♠s > k ❡rs > 1✳ ■ss♦ ❛t❡♥❞❡ ♥♦ss♦ r❡q✉✐s✐t♦✱ ♣♦✐s✱ ❞❡ s > k t❡♠♦s yn > p ❡ ❞❡ rs > 1 t❡♠♦s y > 0✳ ❈♦♠♦ y < r ❝♦♥❝❧✉í♠♦s q✉❡ ❡①✐st❡ ✉♠ y∈M t❛❧ q✉❡ y /∈Dr✳ P♦rt❛♥t♦✱ M 6=Dr✳
✭❜✮ ❙❡ r <0 t♦♠❛♠♦s ✉♠ y > r✱ ♣ór❡♠✱ y <0✳ ❆ss✐♠✱ ❡①✐st❡ ✉♠ y∈Dr t❛❧ q✉❡
y /∈M✳ P♦rt❛♥t♦✱ M 6=Dr✳
❉♦r❛✈❛♥t❡✱ ♣r♦✈❛r❡♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ❈♦rt❡s ❞❡ ❉❡❞❡❦✐♥❞✱ ❝♦♠❡✲ ç❛♥❞♦ ♣❡❧❛ ♣r♦✈❛ ❞❡ ✉♠ ❧❡♠❛ q✉❡ ✉s❛r❡♠♦s ❝♦♠ ♠✉✐t❛ ❢r❡q✉ê♥❝✐❛ ♥❛ ❞❡♠♦♥str❛çã♦ ❞❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s✳
▲❡♠❛ ✶✳✺ ❙❡❥❛ A⊂Q ✉♠ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞✳ ■✳ Q−A={x∈Q|x < a,∀a∈A}❀
■■✳ ❙❡❥❛ x∈Q−A✳ ❙❡ y ∈Q ❡ y≤x✱ ❡♥tã♦ y∈Q−A✳
❉❡♠♦♥str❛çã♦✿ ✭■✮ ❙❡ x∈Q−A❡ a∈A ❡♥tã♦ t❡♠♦s ♣❡❧❛ ❧❡✐ ❞❛ tr✐❝♦t♦♠✐❛ ❞♦s
♥ú♠❡r♦s r❛❝✐♦♥❛✐s q✉❡ ♦✉x > a♦✉x=a♦✉x < a✳ Pr♦✈❛r❡♠♦s q✉❡ s♦♠❡♥t❡ ♦❝♦rr❡
❛ ♣♦ss✐❜✐❧✐❞❛❞❡ x < a✳ ❈♦♠ ❡❢❡✐t♦✱ s❡ x=a∈A✱ ♣♦r ❞❡✜♥✐çã♦✱ ♦ q✉❡ é ✉♠ ❛❜s✉r❞♦❀
s❡ x > a ❡ ❝♦♠♦ a ∈ A ❡ x ∈ Q t❡♠♦s✱ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ■■ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦rt❡✱
q✉❡ x ∈ A✱ ♦ q✉❡ t❛♠❜é♠ é ✉♠ ❛❜s✉r❞♦✳ ❆ss✐♠✱ s♦♠❡♥t❡ ♦❝♦rr❡ x < a✳ P♦rt❛♥t♦✱
Q−A ⊂ {x ∈Q|x < a,∀a ∈ A}✳ ❆❣♦r❛✱ s✉♣♦♥❤❛ q✉❡ y ∈ {x ∈ Q|x < a,∀a ∈ A} ❡ y ∈A✱ ❛ss✐♠✱ t❡rí❛♠♦s y < y✱ ♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✳ ■ss♦ ✐♠♣❧✐❝❛ q✉❡ y ∈Q−A✳
❉❛í ❝♦♥❝❧✉í♠♦s {x∈Q|x < a,∀a ∈A} ⊂Q−A✳ P♦rt❛♥t♦✱ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳
✭■■✮ P❡❧❛ ❤✐♣ót❡s❡✱ t❡♠♦s q✉❡y∈Q✱y≤x < a✱ ♣♦✐s✱x∈A✱ s❡❣✉❡✲s❡ ✐♠❡❞✐❛t❛♠❡♥t❡
q✉❡ y∈Q−A✳
❖ ♣ró①✐♠♦ ❧❡♠❛ ♣r♦✈❛ ❛ ❧❡✐ ❞❛ tr✐❝♦t♦♠✐❛ ♣❛r❛ ♦s ❈♦rt❡s ❞❡ ❉❡❞❡❦✐♥❞✳
▲❡♠❛ ✶✳✻ ❙❡❥❛♠ A, B ∈Q ❈♦rt❡s ❞❡ ❉❡❞❡❦✐♥❞✳ ❊♥tã♦ ♦❝♦rr❡ ❡①❛t❛♠❡♥t❡ ✉♠❛ ❞❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✿ A(B ♦✉ A=B ♦✉ B (A✳
❉❡♠♦♥str❛çã♦✿ ❙❡ A=B✱ ♥ã♦ ❤á ♥❛❞❛ ♦ q✉❡ ♣r♦✈❛r✱ s✉♣♦♥❤❛ q✉❡A6=B✳ ❉❛❞♦ a∈A✱ ❡♥tã♦ t❡♠♦s ❞♦✐s ❝❛s♦s✿
✶✳ ❊①✐st❡ ✉♠ a∈A t❛❧ q✉❡ a /∈B❀ ♦✉
✷✳ ❊①✐st❡ ✉♠ b∈B t❛❧ q✉❡ b /∈A✳
❈❛s♦ ✶✳ ❊♥tã♦ t❡♠♦s a ∈ A ❡ a ∈Q−B✳ P❡❧♦ ▲❡♠❛ ✶✳✺ t❡♠♦s a < b,∀b ∈ B✱
❝♦♠♦ a ∈ A ❡ a < b,∀b ∈ B✱ t❡♠♦s✱ ♣❡❧❛ ♣❛rt❡ ■■ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❈♦rt❡✱ q✉❡
b ∈ A,∀b ∈ B✱ ♦✉ s❡❥❛ ✱ B ⊆ A✳ ❈♦♠♦ s✉♣♦♠♦s A 6= B✱ t❡♠♦s ✱ ♣♦rt❛♥t♦✱ q✉❡ B (A✳ ❖ ❝❛s♦ ✷✱ é s❡♠❡❧❤❛♥t❡ ❛♦ ❝❛s♦ ✶✱ ♣♦rt❛♥t♦✱ t❡♠♦sA(B✳
▲❡♠❛ ✶✳✼ ❙❡❥❛ A ✉♠❛ ❢❛♠í❧✐❛ ❞❡ s✉❜❝♦♥❥✉♥t♦s ♥ã♦✲✈❛③✐♦s ❞❡Q✳ ❙✉♣♦♥❤❛ q✉❡ t♦❞♦
X ∈ A é ✉♠ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞✳ ❙❡ S
X∈AX 6= Q✱ ❡♥tã♦ S
X∈AX é ✉♠ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞✳
❉❡♠♦♥str❛çã♦✿ ❱❛♠♦s ❝❤❛♠❛r B = S
X∈AX✳ ▼♦str❛r❡♠♦s q✉❡ B s❛t✐s❢❛③ ❛s três ♣❛rt❡s ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❈♦rt❡s ❞❡ ❉❡❞❡❦✐♥❞✳
✭■✮ ❈♦♠♦ X 6= ∅✱ ♣❛r❛ t♦❞♦ X ∈ A s❡❣✉❡✲s❡ q✉❡ B = S
X∈AX 6= ∅ ❡✱ ♣❡❧❛ ❤✐♣ót❡s❡✱ B = S
X∈AX 6= Q✳ ❈♦♠ ✐ss♦✱ ♣r♦✈❛♠♦s q✉❡ B s❛t✐s❢❛③ ❛ ♣❛rt❡ ✭■✮ ❞❛ ❞❡✜♥✐çã♦❀
✭■■✮ ❙❡❥❛ x∈B✳ ❙✉♣♦♥❤❛♠♦s ✉♠y ∈Q t❛❧ q✉❡ y≥x✳ ▲♦❣♦✱ x∈X✱ ♣❛r❛ ❛❧❣✉♠ X ∈ A✳ ❖r❛✱ X é ✉♠ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞✱ s❡❣✉❡✲s❡ ❡♥tã♦ ♣❡❧❛ ♣❛rt❡ ✭■■✮ ❞❛
❞❡✜♥✐çã♦ q✉❡y∈X✱ ♦ q✉❡ ✐♠♣❧✐❝❛ y∈B✳ P♦rt❛♥t♦✱ ♣r♦✈❛♠♦s ❛ ♣❛rt❡ ✭■■✮ ❞❛
❞❡✜♥✐çã♦❀
✭■■■✮ ❙❡x∈B✱ ❡♥tã♦x∈X♣❛r❛ ❛❧❣✉♠X ∈ A✳ P❡❧❛ ♣❛rt❡ ✭■■■✮ ❞❛ ❞❡✜♥✐çã♦✱ ❡①✐st❡
y ∈ X✱ q✉❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ y ∈ B✱ t❛❧ q✉❡ y < x✳ P♦rt❛♥t♦✱ ♣r♦✈❛♠♦s ❛
♣❛rt❡ ✭■■■✮ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❈♦rt❡✳
▲❡♠❛ ✶✳✽ ❙❡❥❛♠ A, B ⊆ Q ❈♦rt❡s ❞❡ ❉❡❞❡❦✐♥❞✳ ❊♥tã♦ sã♦ ✈á❧✐❞❛s ❛s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s✿
✭✐✮ ❖ ❝♦♥❥✉♥t♦ ❞❡✜♥✐❞♦ ♣♦rM ={r ∈Q|r=a+b, para algum a∈A e b∈B} é ✉♠ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞❀
✭✐✐✮ ❖ ❝♦♥❥✉♥t♦ ❞❡✜♥✐❞♦ ♣♦r M = {r ∈ Q| −r < c para algum c ∈ Q−A} é
✉♠ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞❀
✭✐✐✐✮ ❙✉♣♦♥❤❛ q✉❡ 0 ∈ Q−A ❡ 0 ∈ Q−B✳ ❖ ❝♦♥❥✉♥t♦ ❞❡✜♥✐❞♦ ♣♦r M = {r ∈
Q|r =ab, para algum a∈A e b∈B} é ✉♠ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞❀
✭✐✈✮ ❙✉♣♦♥❤❛ q✉❡ ❡①✐st❛ ✉♠ q ∈ Q−A t❛❧ q✉❡ q > 0✳ ❖ ❝♦♥❥✉♥t♦ ❞❡✜♥✐❞♦ ♣♦r
M = {r ∈ Q|r > 0 e 1r < c, para algum c ∈ Q−A} é ✉♠ ❈♦rt❡ ❞❡
❉❡❞❡❦✐♥❞✳
❉❡♠♦♥str❛çã♦✿ ❱❛♠♦s ♣r♦✈❛r ❛ ♣❛rt❡ ✭✐✮✿
✭■✮ ❈♦♠ ❡❢❡✐t♦✱ ❝♦♠♦ A ❡ B sã♦ ♥ã♦✲✈❛③✐♦s ❡①✐st❡♠ a ∈ A ❡ b ∈ B✳ ▲♦❣♦✱ ❡①✐st❡ r =a+b∈M✳ ❉❛í✱ t❡♠♦s M 6=∅✳ ❆❣♦r❛✱ ♣❡❧❛ ❤✐♣ót❡s❡ ❞❡ ❈♦rt❡✱ t❡♠♦s q✉❡
A6=Q ❡ B 6=Q✳ ❆ss✐♠✱ ❡①✐st❡♠ p∈Q−A ❡ q∈Q−B q✉❡✱ ♣❡❧♦ ▲❡♠❛ ✶✳✺✱
sã♦ t❛✐s q✉❡ p < a,∀a ∈ A ❡ q < b,∀b ∈ B❀ s♦♠❛♥❞♦ ❛s ❞✉❛s ❞❡s✐❣✉❛❧❞❛❞❡s
❛♥t❡r✐♦r❡s✱ t❡r❡♠♦sp+q < a+b,∀a+b ∈M✳ ❉❛í✱ t❡♠♦s q✉❡p+q∈Q−M✱✐st♦
é✱ M 6=Q✳ P♦rt❛♥t♦✱ ♣r♦✈❛♠♦s q✉❡ M 6=∅ ❡ M 6=Q✳
✭■■✮ ❙✉♣♦♥❤❛ ✉♠ r ∈ M ❡ ✉♠ s ∈ Q t❛✐s q✉❡ s ≥ r✳ ❈♦♠♦ r ∈ M t❡♠♦s r = a+b✱ ♣❛r❛ ❛❧❣✉♠ a ∈ A ❡ b ∈ B✳ ❯s❛♥❞♦ ❛ ❧❡✐ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❞✐t✐✈❛
♣❛r❛ ♦s r❛❝✐♦♥❛✐s t❡♠♦s a + (s− r) ≥ a✱ ❛ss✐♠✱ ♣❡❧❛ ✷❛ ♣❛rt❡ ❞❛ ❞❡✜♥✐çã♦✱ a+ (s−r)∈A✳ ❉❛í✱ s❡❣✉❡✲s❡ q✉❡ s= [a+ (s−r)] +b ∈M✳
✭■■■✮ ❙❡ r ∈ M✱ ❡♥tã♦ ❡①✐st❡♠ a ∈ A ❡ b ∈ B t❛✐s q✉❡ r = a+b✳ ❈♦♠♦ A é
✉♠ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞ ❡①✐st❡ ✉♠ c ∈ A t❛❧ q✉❡ c < a✳ ❉❛í✱ s❡❣✉❡✲s❡ q✉❡ s=c+b < a+b =r✱ ♦♥❞❡c∈A ❡b ∈B✳ ❆ss✐♠✱ s < r ❡✱ ♣♦rt❛♥t♦✱ s∈M✳
P❛rt❡ ✭✐✐✮✿
✭■✮ ❙❛❜❡♠♦s q✉❡ A 6= Q✱ ❧♦❣♦✱ ❡①✐st❡ ✉♠ b ∈ Q −A✳ P❡❧❛ s❡❣✉♥❞❛ ♣❛rt❡ ❞♦
▲❡♠❛ ✶✳✺✱ t❡♠♦s q✉❡ b − 1 ∈ Q −A✳ ◆♦t❡ q✉❡ −[−(b − 1)] < b✱ ✐st♦ é✱
−(b−1)∈ M✳ ❆ss✐♠✱ ✜❝❛ ♣r♦✈❛❞♦ q✉❡M 6=∅✳ ❱❛♠♦s ♣r♦✈❛r q✉❡ −a /∈M✱
✐st♦ é✱ −a ∈ Q−M✳ ❘❡s✉❧t❛♥❞♦ ❞✐ss♦✱ ♦ ❢❛t♦ ❞❡ q✉❡ M 6= Q✳ ❈♦♠ ❡❢❡✐t♦✱
♥♦t❡ q✉❡ −(−a) ∈A✱ ✐st♦ é✱ −(−a)∈/ Q−A✳ ❖r❛✱ ♣❡❧❛ ♣❛rt❡ ✷ ❞♦ ▲❡♠❛ ✶✳✺
✈❡♠♦s q✉❡ ❡①✐st❡ ✉♠ b ∈ Q−A t❛❧ q✉❡ −(−a) > b✳ ■ss♦ ✐♠♣❧✐❝❛ −a /∈ M✳
P♦rt❛♥t♦✱ s❡❣✉❡✲s❡ ♦ r❡s✉❧t❛❞♦✳
✭■■✮ ❙✉♣♦♥❤❛ q✉❡x∈M ❡ ♣❛r❛y∈Qt❡♠♦s y≥x✳ ❆ss✐♠✱ ♣❛r❛ ❛❧❣✉♠c∈Q−A
t❡♠♦s −x < c✱ ♦ q✉❡ ✐♠♣❧✐❝❛ −y≥ −x < c✳ P♦rt❛♥t♦✱ y∈M✳
✭■■■✮ ❙❡ x ∈ M✱ ❡♥tã♦ ❡①✐st❡ ✉♠ c ∈ Q−A t❛❧ q✉❡ −x < c ♦ q✉❡ ✐♠♣❧✐❝❛ x >
−c✳ ❈❤❛♠❛♥❞♦ ❞❡ y= x+(2−c)✱ t❡♠♦s q✉❡✱ ✉s❛♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦s ♥ú♠❡r♦s
r❛❝✐♦♥❛✐s✱ −c < x+(2−c) < x✳ ❉❡ss❛ ❞❡s✐❣✉❛❧❞❛❞❡✱ ♦❜t❡♠♦s y < x ❡ −x <
−(x+(−c))
2 < c✱ ♦ q✉❡ ♥♦s ❧❡✈❛ ❛ ❝♦♥❝❧✉✐r q✉❡ y∈M✳
P❛rt❡ ✭✐✐✐✮
✭■✮ ❈♦♠♦ A 6= ∅ ❡ B 6= ∅✱ ❡①✐st❡♠ a ∈ A ❡ b ∈ B✳ ❆ss✐♠✱ ❡①✐st❡ r t❛❧ q✉❡ r = ab✳ ❖r❛✱ r ∈ M✱ ✐st♦ é✱ M 6= ∅✳ ❈❧❛r❛♠❡♥t❡✱ 0 ∈/ M✱ ♣♦✐s✱ ♣♦❞❡♠♦s
❡s❝r❡✈❡r0 = a·0 ❝♦♠a ∈A ❡0∈/ B✱ ♣♦r ❤✐♣ót❡s❡✳ ❆ss✐♠✱ M 6=Q✳ P♦rt❛♥t♦✱ ♣r♦✈❛♠♦s q✉❡ M 6=∅ ❡ M 6=Q✳
✭■■✮ ❙✉♣♦♥❤❛ x ∈ M ❡ y ∈ Q t❛✐s q✉❡ y ≥ x✳ ❙❡ x ∈ M✱ ❡♥tã♦ ❡①✐st❡♠ a ∈ A ❡ b ∈ B t❛✐s q✉❡ x = ab✳ P❡❧❛ ❤✐♣ót❡s❡✱ ✈❡♠♦s q✉❡ a > 0 ❡ b > 0 ✳ ❋❛③❡♥❞♦ y=a·ya✱ ❝♦♥❝❧✉í♠♦s q✉❡a∈A❡ ya ∈B✳ ❆ss✐♠✱ ✈❡♠♦s q✉❡y∈M✳ P♦rt❛♥t♦✱
t❡♠♦s q✉❡ s❡ y≥x✱ ❡♥tã♦ y ∈M✳
✭■■■✮ ❙❡ x ∈ M t❡♠♦s q✉❡ ❡①✐st❡♠ a ∈ A ❡ b ∈ B t❛✐s q✉❡ x = ab✳ P❡❧❛ ❤✐♣ót❡s❡✱
❞❡❞✉③✐♠♦s q✉❡ a > 0 ❡ b > 0 ✳ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞✭■■■✮✱
❡①✐st❡♠ c∈ A ❡ d ∈B t❛✐s q✉❡ c < a ❡ d < b✳ ■ss♦ ✐♠♣❧✐❝❛ ❡♠y =cd < ab ❡ y∈M✳ P♦rt❛♥t♦✱ ❡①✐st❡ ✉♠ y∈M t❛❧ q✉❡ y < x✳
P❛rt❡ ✭✐✈✮
✭■✮ P❡❧❛ ❤✐♣ót❡s❡✱q >0❡q ∈Q−A✱ ✐st♦ é✱ 0< q < a✱ ♣❛r❛ t♦❞♦a ∈A✳ ❙❡r∈Q é t❛❧ q✉❡ r ≤0✱ ❡♥tã♦ r /∈M ♦ q✉❡ ❧❡✈❛ ❛ ❝♦♥❝❧✉✐r q✉❡ M 6=Q✳ ❙❛❜❡♠♦s q✉❡ 0 < q2 < q✳ ❆ss✐♠ s❡ t♦♠❛r♠♦s r = 2q−1 t❡r❡♠♦s r ∈ M ❞♦ q✉❛❧ r❡s✉❧t❛ M 6=∅✳ P♦rt❛♥t♦✱ ♣r♦✈❛♠♦s q✉❡M 6=∅ ❡M 6=Q✳
✭■■✮ ❙❡ x∈M ❡ y∈Q sã♦ t❛✐s q✉❡ y≥x✳ ❉❡ x∈M ❡ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡✱ ♦❜t❡♠♦s 1
y <
1
x < c ♣❛r❛ ❛❧❣✉♠ c∈Q−A✳ P♦rt❛♥t♦✱ y∈M✳
✭■■■✮ ❙❡ x ∈ M✱ ❡♥tã♦ x > 0 ❡ ♣❛r❛ ❛❧❣✉♠ c ∈ Q− A t❡♠♦s 0 < 1
x < c ❝✉❥❛ ❞❡s✐❣✉❛❧❞❛❞❡ r❡s✉❧t❛ ❡♠ 1
c < x✳ P♦❞❡♠♦s t♦♠❛r ✉♠y= x+1
c
2 ✳ ❆ss✐♠✱ ♣♦❞❡♠♦s
❝♦♥❝❧✉✐r q✉❡y∈Q ❡ 1c < y < x✳ ❉❡ss❛ ú❧t✐♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ r❡s✉❧t❛0< 1
y < c✱ ✐st♦ é✱ y∈M✳ P♦rt❛♥t♦✱ s❡x∈M ❡①✐st❡ ✉♠ y∈M t❛❧ q✉❡ y < x✳
❖ r❡s✉❧t❛❞♦ ❞❛ s❡❣✉♥❞❛ ♣❛rt❡ ❞♦ ❧❡♠❛ ❛ s❡❣✉✐r s❡rá ✉s❛❞♦ ♥❛ ♣r♦✈❛ ❞♦ ❚❡♦r❡♠❛ ✷✳✸✱ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡ ❡♠ s✉❛ ♣❛rt❡ ✭✺✮✱ q✉❡ ♠♦str❛ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s r❡❧❛t✐✈❛s ❛♦s ♥ú♠❡r♦s r❡❛✐s❀ ❛ s✉❛ ♣r✐♠❡✐r❛ ♣❛rt❡✱ s❡r✈✐rá ❞❡ ❜❛s❡ ♣❛r❛ ♣r♦✈❛ ❞❛ s✉❛ s❡❣✉♥❞❛ ♣❛rt❡✱ ❡ t❛♠❜é♠✱ ❞❛ ♣❛rt❡ ✭✹✮ ❞♦ ❚❡♦r❡♠❛ ✷✳✶✳ P❛r❛ ♣r♦✈❛r♠♦s ♦ ❧❡♠❛ ❛ s❡❣✉✐r✱ ❡♥✉♥❝✐❛r❡♠♦s ❞✉❛s ♣r♦♣♦s✐çõ❡s s♦❜r❡ ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✱ s❡♠ ❞❡♠♦♥strá✲❧❛s✱ q✉❡ s❡rã♦ ✉s❛❞❛s ♥❛ ♣r♦✈❛ ❞♦ ❧❡♠❛ q✉❡ s❡ s❡❣✉❡✳
Pr♦♣♦s✐çã♦ ✶✳✸ ❙❡❥❛♠ r, s ∈ Q r❛❝✐♦♥❛✐s t❛✐s r > 0 ❡ s > 0✳ ❊♥tã♦ ❡①✐st❡ ✉♠
♥❛t✉r❛❧ n∈N t❛❧ q✉❡ s❡ t❡♥❤❛ s < nr✳
❚❡♦r❡♠❛ ✶✳✶ ✭Pr✐♥❝í♣✐♦ ❞❛ ❇♦❛ ❖r❞❡♥❛çã♦✮ ❙❡❥❛A⊆N✉♠ s✉❜❝♦♥❥✉♥t♦ ♥ã♦✲ ✈❛③✐♦✳ ❊♥tã♦ ❡①✐st❡ ✉♠ m∈A t❛❧ q✉❡ m≤x✱ ♣❛r❛ t♦❞♦ x∈A✳
▲❡♠❛ ✶✳✾ ❙❡❥❛ A ✉♠ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞ ❡ y∈Q✳
✭■✮ ❙✉♣♦♥❞♦ q✉❡ y > 0✳ ❊♥tã♦ ❡①✐st❡♠ u ∈A ❡ v ∈ Q−A t❛✐s q✉❡ y =u−v✱ ❡ v < e✱ ♣❛r❛ ❛❧❣✉♠ e∈Q−A ❀
✭■■✮ ❙✉♣♦♥❤❛ q✉❡ y > 1 ❡ q✉❡ ❡①✐st❛ ✉♠ q ∈ Q−A t❛❧ q✉❡ q >0✳ ❊♥tã♦ ❡①✐st❡♠ r ∈A❡ s∈Q−A t❛✐s q✉❡s >0 ❡y > r
s ✱ ❝♦♠ s < g✱ ♣❛r❛ ❛❧❣✉♠g ∈Q−A ✳
❉❡♠♦♥str❛çã♦✿
✭■✮ ❈♦♠♦ Aé ✉♠ ❈♦rt❡ t❡♠♦s✱ A6=∅❡A 6=Q✳ ■ss♦ ✐♠♣❧✐❝❛ q✉❡ ❡①✐st❡♠ z ∈A❡ w ∈Q−A✳ ❙❛❜❡♠♦s✱ ♣❡❧♦ ▲❡♠❛ ✶✳✺✱ q✉❡ w < z ♦ q✉❡ ❧❡✈❛z−w >0✳ ❈♦♠♦ z−w❡y sã♦ r❛❝✐♦♥❛✐s ♣♦s✐t✐✈♦s ❡①✐st❡ ✉♠n∈N t❛❧ q✉❡z−w < ny✳ ❉❛ q✉❛❧
t✐r❛♠♦s ❡ r❡❡s❝r❡✈❡♠♦sz+n(−y)< w✳ ❉♦ ▲❡♠❛ ✶✳✺ t✐r❛♠♦sz+n(−y)∈Q−A✳
❱❛♠♦s ❞❡✜♥✐r ♦ ❝♦♥❥✉♥t♦ M t❛❧ q✉❡✿
M ={p∈N|z+p(−y)∈Q−A}.
❆♥t❡r✐♦r♠❡♥t❡✱ ❝♦♠♦ ❡①✐st❡ n ∈ N t❛❧ q✉❡ z+n(−y) ∈ Q−A s❡❣✉❡✲s❡ q✉❡ M 6= ∅✳ ❖r❛✱ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ❇♦❛ ❖r❞❡♥❛çã♦✱❡①✐st❡ ✉♠ m ∈ N t❛❧ q✉❡
z+m(−y)∈Q−A ❡z+ (m−1)(−y)∈A✳ ❉✐ss♦ r❡s✉❧t❛ ♣❡❧♦ ▲❡♠❛ ✶✳✺ ✭■■✮
q✉❡✱ ❡①✐st❡ ✉♠e∈Q−At❛❧ q✉❡z+m(−y)< e✳ ❋❛ç❛♠♦su=z+(m−1)(−y)
❡v =z+m(−y)✳ ❉♦ q✉❛❧ t✐r❛♠♦sy=u−v✱ ❝♦♠v < e✱ ♣❛r❛ ❛❧❣✉♠e∈Q−A✳
✭■■✮ ❆ ❞❡♠♦♥str❛çã♦ ❞❡ss❛ ✷❛ ♣❛rt❡ ❞♦ ▲❡♠❛ ❡①✐❣❡ ✉♠ ❛rt✐❢í❝✐♦ ❡♥❣❡♥❤♦s♦✳ ❱❛♠♦s
♠♦str❛r ❛ ❝♦♥str✉çã♦ ❞♦ r❛❝✐♦❝í♥✐♦ ❡♠ ❞✉❛s ❡t❛♣❛s✳ Pr✐♠❡✐r❛ ❊t❛♣❛ ❞♦ ❘❛❝✐♦✲ ❝í♥✐♦✿ ➱ ♥❛t✉r❛❧ q✉❡ t❡♥❤❛♠♦s ✐♥❝❧✐♥❛çã♦ ❡♠ ✉s❛r ❛ ✶❛♣❛rt❡ ❞❡ss❡ ♠❡s♠♦ ❧❡♠❛
♣❛r❛ ❛♣❛r❡❝❡r ❛s ❞✐t❛s ❧❡tr❛s ✉ ❡ ✈ q✉❡ ❛♣❛r❡❝❡♠ ❝♦♠♦ ❢♦r♠❛ ❞❡ q✉♦❝✐❡♥t❡ ❝♦♠
y > u
v✱♦♥❞❡ u∈ A✱ ❡ v ∈Q−A✱ ❝♦♠ v >0 ❡ v < g✱ ♣❛r❛ ❛❧❣✉♠ g ∈ Q−A✳ ❆ ✐❞❡✐❛ q✉❡ s❡ t❡♠ é ❝♦♥str✉✐r ✉♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ (y−1)v > u−v ✭❝✉❥❛ ✐♠✲
♣❧✐❝❛çã♦ ♥♦s ❞á y > u
v✮✳ P❛r❛ q✉❡ ✐ss♦ ♦❝♦rr❛✱ é ♥❡❝❡ssár✐♦ q✉❡ ❡❧❛ ❛❞✈❡♥❤❛ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡(y−1)v >(y−1)q
2 =u−v✳ ❆❣♦r❛✱ ♣❛r❛ q✉❡ ❡ss❛ ❞❡s✐❣✉❛❧❞❛❞❡✱
❥✉♥t♦ ❝♦♠ ❡ss❛ ✐❣✉❛❧❞❛❞❡✱ t❡♥❤❛ s❡♥t✐❞♦ ❞❡ s❡r✱ ✐♠♣♦♠♦s ♥❛ ❤✐♣ót❡s❡ ❞♦ ❧❡♠❛
y > 1 ❡ ❡①✐st❛ ✉♠ q > 0✱♦♥❞❡ q ∈ Q −A ♣❛r❛ q✉❡ ❢❛ç❛ s❡♥t✐❞♦ ♦ ✉s♦ ❞❛
♣❛rt❡ ✭■✮ ❞♦ ❧❡♠❛✳ ❖✉tr♦ ✐♠♣❛ss❡ q✉❡ ❞❡✈❡♠♦s r❡s♦❧✈❡r é ❛ r❡❧❛çã♦ ❡♥tr❡ v ❡
q
2✳ ❙❡ ❞✉r❛♥t❡ ❛ ♣r♦✈❛ ✐♠♣♦♠♦s v >
q
2 > 0✱ ❡ss❛ ❡t❛♣❛ ❞❛ ❞❡♠♦♥str❛çã♦ ✜❝❛
r❡s♦❧✈✐❞❛✳ ❱❡❥❛♠♦s✿ ❈♦♠♦y >1❡q >0t❡♠♦s(y−1)q2 >0✳ ▲♦❣♦✱ ♣❡❧❛ ♣❛rt❡
✭■✮ ❞❡ss❡ ❧❡♠❛ t❡♠♦s q✉❡ ❡①✐st❡♠ u∈A❡v ∈Q−A t❛✐s q✉❡(y−1)q
2 =u−v
❝♦♠ v < g✱ ♣❛r❛ ❛❧❣✉♠ g ∈ Q−A✳ ❙❡ ❡st❡ v ❢♦r t❛❧ q✉❡ v > q2✱ ❡♥tã♦ t❡r❡✲
♠♦s (y−1)v > (y−1) q2 = u−v✱ ♦✉ s❡❥❛✱ (y−1)v > u−v ♦ q✉❡ ✐♠♣❧✐❝❛
s✉❝❡ss✐✈❛♠❡♥t❡ ❡♠ yv−v > u−v ❡ y > uv✱ ♣♦✐s✱ v > 0✳ ❈❤❛♠❛♥❞♦ r = u ❡ s=v t❡♠♦sy > r
s✱ ❝♦♠s >0 ❡s < g ∈Q−A✳ P♦rt❛♥t♦✱ ♣r♦✈❛♠♦s q✉❡ ❡①✐s✲ t❡♠r∈A❡s∈Q−At❛✐s q✉❡y > r
s ❝♦♠s >0❡s < g✱♣❛r❛ ❛❧❣✉♠g ∈Q−A✳
❆❣♦r❛✱ ✈❛♠♦s ❝♦♥str✉✐r ♦ r❛❝✐♦❝í♥✐♦ q✉❛♥❞♦ v ≤ q
2 ✳ ❆ ✐❞❡✐❛ ❝♦♥s✐st❡ ❡♠
❝♦♥str✉✐r ♦✉tr♦s ❡❧❡♠❡♥t♦s r ∈ A ❡ s ∈ Q− A t❛✐s q✉❡ ❛✐♥❞❛ t❡♥❤❛♠♦s (y−1)q2 =u−v =r−s ❞❡✈❡♠♦s ❢❛③❡r q✉❡ ♦❝♦rr❛(y−1)s >(y−1)q2 =r−s
❡ q✉❡ s = 34q✳ ❆❣♦r❛✱ ❜❛st❛ ♠♦str❛r♠♦s ❝♦♠ ✐ss♦ q✉❡ r ∈ A ❡ s ∈ Q−A ✳
❈♦♠ ❡❢❡✐t♦✱ é ✐♠❡❞✐❛t♦ q✉❡ ♣♦r s = 3 4 q <
1
2q < q r❡s✉❧t❛ s ∈ Q−A✱ ❡ q✉❡
♣♦r r = u+ (s−v) > u ✭♣♦✐s✱ s > q2 ≥ v✮ r❡s✉❧t❛ r ∈ A✳ ❆ss✐♠✱ ❛ s❡❣✉♥❞❛
♣❛rt❡ ❞♦ ❧❡♠❛ ✜❝❛ ♣r♦✈❛❞♦✳ ❱❡❥❛♠♦s✿ ❋❛ç❛♠♦s r=u+ (s−v) ❡ s❡❥❛ s= 34q✳
❆ ♣❛rt✐r ❞✐ss♦ t❡♠♦s r−s =u−v = (y−1)q
2 <(y−1)s ❡✱ ♣♦rt❛♥t♦✱ r < sy
♦ q✉❡ ✐♠♣❧✐❝❛ y > r
s✳ ❆❣♦r❛✱ ❝♦♠♦ s =
3
4q < q ❡ q ∈ Q−A ❝♦♥❝❧✉í♠♦s q✉❡ s ∈ Q−A ❡ q✉❡✱ ♣♦r tr❛♥s✐t✐✈✐❞❛❞❡✱ s < q = g ∈ Q−A✳ ❖r❛✱ ✈❡♠♦s q✉❡ r =u+ (s−v) > u✱ ♣♦✐s✱ s−v > 0✳ ▲♦❣♦✱ r ∈ A✳ P♦rt❛♥t♦✱ ♣r♦✈❛♠♦s q✉❡
❡①✐st❡♠ r∈A ❡s ∈Q−At❛✐s q✉❡ y > r
s ❝♦♠ s < g✱ ♣❛r❛ ❛❧❣✉♠ g ∈Q−A ✳
❈❛♣ít✉❧♦ ✷
❈♦♥str✉çã♦ ❞♦s ◆ú♠❡r♦s ❘❡❛✐s
◆♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r✱ t✐✈❡♠♦s ♦ ❞✉r♦ tr❛❜❛❧❤♦ ❞❡ ❛ss✐♠✐❧❛r ❛ ❞❡✜♥✐çã♦ ❞❡ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞ ❡ ❛ t❛r❡❢❛ ár❞✉❛ ❞❡ ♣r♦✈❛r ♦s ❧❡♠❛s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞♦s ♥❡st❡ ❝❛♣ít✉❧♦✳ ❊st❡ ❝❛♣ít✉❧♦ ♠♦str❛✱ ❞❡ ❢❛t♦✱ ❛ ❝♦♥str✉çã♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ✉s❛♥❞♦ ♦s ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞✳ ❱❛♠♦s ✈❡r✐✜❝❛r q✉❡✱ ❝♦♠ ❛ ❞❡✜♥✐çã♦ s❡❣✉✐♥t❡✱ ♦s ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞ é ✐❞❡♥t✐✜❝❛❞♦ ❝♦♠ ♦s ◆ú♠❡r♦s ❘❡❛✐s ❡✱ ♠❛✐s ❛❞✐❛♥t❡✱ s❡ ❝♦♠♣♦rt❛♠ ❞❛ ♠❡s♠❛ ♠❛♥❡✐r❛ q✉❡ ♦s ♥ú♠❡r♦s r❡❛✐s ❡♠ r❡❧❛çã♦ ❛s s✉❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦✱ ♠✉❧t✐♣❧✐❝❛çã♦✱ ✐♥✈❡rs♦ ❛❞✐t✐✈♦ ❡ ♠✉❧t✐♣❧✐❝❛t✐✈♦✱ ❡ ❛ r❡❧❛çã♦ ❞❡ ♦r❞❡♠✳
❉❡✜♥✐çã♦ ✷✳✶ ❖ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✱ ❞❡♥♦t❛❞♦ ♣♦r ℜ✱ é ❞❡✜♥✐❞♦ ♣♦r✿
ℜ={A ⊆Q|A seja um corte de Dedekind}.
❙❛❜❡♠♦s q✉❡ ♦s ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞ sã♦ s✉❜❝♦♥❥✉♥t♦s ❞❡ ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✱ ❞❡✜♥✐✲ r❡♠♦s ❛ r❡❧❛çã♦ ❞❡ ❖r❞❡♠ s♦❜r❡ ♦s ◆ú♠❡r♦s r❡❛✐s ❡♠ t❡r♠♦s ❞❛ r❡❧❛çã♦ ❞❡ ✧❡stá ❝♦♥t✐❞♦✧s♦❜r❡ ❝♦♥❥✉♥t♦ ❞❡ ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✳
❉❡✜♥✐çã♦ ✷✳✷ ❉❛❞♦s A, B ∈ ℜ✱ ❞✐③❡♠♦s q✉❡ ✧A é ♠❡♥♦r ❞♦ q✉❡ B✧✱ ❞❡♥♦t❛❞♦ ♣♦r
A < B s❡✱ B ⊆ A ❡ A =6 B ✭B ( A✮✱ ✐st♦ é✱ q✉❛♥❞♦ A é ✉♠ s✉❜❝♦♥❥✉♥t♦ ♣ró♣r✐♦
❞❡ B✳ ❉❡✜♥✐♠♦s A ≤ B✱ ✐st♦ é✱ ✧A é ♠❡♥♦r ❞♦ q✉❡ ♦✉ ✐❣✉❛❧ ❛ B✧s❡✱ B ⊂ A ♦✉ A=B✭B ⊆A✮✳
❯s❛r❡♠♦s ♦s ✐t❡♥s ✭✐✮ ❡ ✭✐✐✮ ❞♦ ▲❡♠❛ ✶✳✽ ♣❛r❛ ❞❡✜♥✐r♠♦s ❛ ❛❞✐çã♦ ❡ ❡❧❡♠❡♥t♦ ♦♣♦st♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s✳ ➱ ❢á❝✐❧ ✈❡r✱ ♣❡❧♦ ♠❡s♠♦ ❧❡♠❛✱ q✉❡ ❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ❡❧❡♠❡♥t♦ ♦♣♦st♦ sã♦ ❢❡❝❤❛❞❛s s♦❜r❡ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✳
❉❡✜♥✐çã♦ ✷✳✸ ❉❛❞♦sA, B ∈ ℜ✱ ❞❡✜♥✐♠♦s ❛ s♦♠❛ ❞❡ A❝♦♠B✱ ❞❡♥♦t❛❞❛ ♣♦rA+B
♦ ♥ú♠❡r♦ r❡❛❧
A+B ={r ∈Q|r=a+b para algum a∈A e b∈B}.
❉❡✜♥✐çã♦ ✷✳✹ ❉❛❞♦ A ∈ ℜ ✱ ❞❡✜♥✐♠♦s ♦♣♦st♦ ❞❡ A✱ ❞❡♥♦t❛❞♦ ♣♦r −A ♦ ♥ú♠❡r♦
r❡❛❧
−A ={r ∈Q| −r < c para algum c∈Q−A}.
❆ ❞❡✜♥✐çã♦ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ✐♥✈❡rs♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ ♣❛r❛ ♦s ♥ú♠❡r♦s r❡❛✐s é ✉♠ ♣♦✉❝♦ ♠❛✐s ❝♦♠♣❧✐❝❛❞❛ ❞❡ q✉❡ ❛ ❞❡✜♥✐çã♦ ❞❡ ❛❞✐çã♦ ❡ ❡❧❡♠❡♥t♦ ♦♣♦st♦✱ ♣♦✐s✱ ♥❡❧❡ ✈❛♠♦s ♣r❡❝✐s❛r ❞❡ ❛❧❣✉♥s r❡q✉✐s✐t♦s✳ ❈♦♠❡ç❛r❡♠♦s ❝♦♠ ♦ s❡❣✉✐♥t❡ ❧❡♠❛✳
▲❡♠❛ ✷✳✶ ❙❡❥❛ A∈ ℜ✱ ❡ r∈Q✳ ❊♥tã♦ sã♦ ✈á❧✐❞❛s ❛s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s✿ ✭✐✮ A > Dr s❡ ❡ s♦♠❡♥t❡ s❡ ❡①✐st❡q ∈Q−A t❛❧ q✉❡ q > r✳
✭✐✐✮ A≥Dr s❡ ❡ s♦♠❡♥t❡ s❡ r∈Q−A s❡ ❡ s♦♠❡♥t❡ s❡a > r✱ ♣❛r❛ t♦❞♦ a∈A✳ ✭✐✐✐✮ ❙❡ A < D0 ❡♥tã♦ −A≥D0 ✳
❉❡♠♦♥str❛çã♦✿
✭✐✮ P♦r ❤✐♣ót❡s❡✱ t❡♠♦sA∈Dr ❡A=6 Dr✱ ✐st♦ q✉❡r ❞✐③❡r q✉❡ r < a,∀a∈A❡ q✉❡ ❡①✐st❡ ✉♠ q ∈ Dr t❛❧ q✉❡ q /∈ A ♦✉✱ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ ❡①✐st❡ ✉♠ q ∈ Q−A t❛❧ q✉❡ r < q✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ ❡①✐st❡ q∈Q−A t❛❧ q✉❡ r < q✱ t❡♠♦s ♣❡❧♦
❧❡♠❛ ✶✳✼✱ q✉❡ r < q≤a,∀a∈A✳ ❆ss✐♠ ✈❡♠♦s q✉❡ A ⊆Dr✳ ❙❡ t♦♠❛r♠♦s ✉♠ y= r+2q ✈❡♠♦s ❞❡ r < y < q q✉❡ y∈Dr ❡ y∈Q−A ✳▲♦❣♦✱ ❡①✐st❡ ✉♠ y∈Dr t❛❧ q✉❡ y /∈A✳ P♦rt❛♥t♦✱ A 6=Dr✳
✭✐✐✮ A ⊆ Dr ♦✉ A =Dr é ❡q✉✐✈❛❧❡♥t❡✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ s✉❜❝♦♥❥✉♥t♦ ♦✉ ❞❛ ✐❣✉❛❧✲ ❞❛❞❡ ❞❡ ❝♦♥❥✉♥t♦s✱ ❛ r < a,∀a ∈A✱ ❡ q✉❡✱ ♣♦r s✉❛ ✈❡③✱ ♣❡❧❛ ♣❛rt❡ ✭■✮ ❞♦ ❧❡♠❛
✶✳✺✱ é ❡q✉✐✈❛❧❡♥t❡ ❛ r∈Q−A ✳
✭✐✐✐✮ ❙✉♣♦♥❤❛ q✉❡ A < D0✳ ❉✐ss♦ r❡s✉❧t❛ q✉❡ ❡①✐st❡ ✉♠ q ∈A t❛❧ q✉❡ q ≤0✳ P❡❧❛
♣❛rt❡ ✭■■✮ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞✱ ❝♦♥❝❧✉í♠♦s q✉❡ 0 ∈ A✳ ❆ss✐♠✱ 0 ∈/ Q−A✱ ❡ ♣♦rt❛♥t♦ −0 ∈/ Q−A✱ ♦ q✉❡ ✐♠♣❧✐❝❛ 0 ∈ −/ A✳ P❡❧❛ ♣❛rt❡ ✭■■✮
❞❡st❡ ❧❡♠❛ ❝♦♥❝❧✉í♠♦s q✉❡ −A≥D0 ♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✳
❆❣♦r❛✱ ❛s ❞❡✜♥✐çõ❡s ❛ s❡❣✉✐r s♦❜r❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ❡❧❡♠❡♥t♦ ✐♥✈❡rs♦ ✭✐♥✈❡rs♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✮ s♦❜r❡ ♥ú♠❡r♦s r❡❛✐s ❢❛③❡♠ s❡♥t✐❞♦ ❞❡✈✐❞♦ ❛♦s ▲❡♠❛s ✶✳✽ ❡ ✷✳✶✳
❉❡✜♥✐çã♦ ✷✳✺ ❙❡❥❛♠ A, B ∈ ℜ✳ ❉❡✜♥✐♠♦s ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡A ♣♦r B✱ ♦ q✉❛❧ ❞❡♥♦✲
t❛♠♦s ♣♦r A·B✱ ♦ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞✿
✭■✮ A·B ={r∈Q|r =ab para algum a∈A e b∈B}✱ s❡ A≥D0 ❡ B ≥D0✳
✭■■✮ A·B =−[(−A)·B]✱ s❡ A < D0 ❡ B ≥D0 ✳
✭■■■✮ A·B =−[A·(−B)]✱ s❡ A≥D0 ❡ B < D0✳
✭■❱✮ A·B = (−A)·(−B)✱ s❡ A < D0 ❡ B < D0 ✳
❉❡✜♥✐çã♦ ✷✳✻ ❉❛❞♦ A ∈ ℜ✳ ❉❡✜♥✐♠♦s ♦ ❡❧❡♠❡♥t♦ ✐♥✈❡rs♦ ❞❡ A✱ ❞❡♥♦t❛❞♦ ♣♦r A−1✱ ♦ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞✿
✭■✮ A−1 ={r∈Q|r >0 e 1
r < c para algum c∈Q−A}✱ s❡ A < D0✳ ✭■■✮ A−1 =−(−A)−1✱ s❡ A > D0✳
❈♦♠ ❛s ❞❡✜♥✐çõ❡s ❞❛s ♦♣❡r❛çõ❡s ❜ás✐❝❛s ❡♠ ♠ã♦s✱ ❡s❜♦ç❛r❡♠♦s ❡ ♣r♦✈❛r❡♠♦s✱ ❛ s❡❣✉✐r✱ ❛s ♣r♦♣r✐❡❞❛❞❡s ❛❧❣é❜r✐❝❛s ❢✉♥❞❛♠❡♥t❛✐s ❞♦s ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞✳ ❋r✐s❡♠♦s q✉❡✱ ❛s ♣r♦♣r✐❡❞❛❞❡s ❡stã♦ ❡♥✉♠❡r❛❞❛s ❡ ♦r❣❛♥✐③❛❞❛s ♥✉♠❛ s❡q✉ê♥❝✐❛ ❧ó❣✐❝❛ ❞❡ ❢♦r♠❛ q✉❡ ❛ ♣r♦✈❛ ❞❡ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡♣❡♥❞❡ ❞❛ ❛❝❡✐t❛çã♦ ❞❛ ♣r♦♣r✐❡❞❛❞❡ ❛♥t❡r✐♦r ❥á ❞❡♠♦♥str❛❞❛✳
❚❡♦r❡♠❛ ✷✳✶ ❙❡❥❛♠ A, B, C ∈ ℜ✳ ❊♥tã♦ sã♦ ✈á❧✐❞❛s ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s ❆❞✐t✐✈❛s✿
✶✳ A+ (B+C) = (A+B) +C ✭▲❡✐ ❆ss♦❝✐❛t✐✈❛ ♣❛r❛ ❆❞✐çã♦✮
✷✳ A+B =B+A ✭▲❡✐ ❈♦♠✉t❛t✐✈❛✮
✸✳ A+D0 =A ✭▲❡✐ ❞❛ ■❞❡♥t✐❞❛❞❡ ❆❞✐t✐✈❛✮
✹✳ A+ (−A) =D0 ✭▲❡✐ ❞♦ ■♥✈❡rs♦ ❆❞✐t✐✈♦✮
❉❡♠♦♥str❛çã♦✿
✶✳ ❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ s♦♠❛ ❞❡ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞ t❡♠♦s✿
A+ (B+C) ={r∈Q|r =a+ (b+c), para algum a ∈A, b∈B, e c∈C}
={r∈Q|r= (a+b)+c, para algum a∈A, b ∈B, e c∈C}= (A+B)+C.
✷✳ ❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ s♦♠❛ ✈❡♠♦s q✉❡A+B ={r∈Q|r=a+b para algum a∈
A e b∈B}={r∈Q|r =b+a para algum b∈B e a∈A}=B+A✳
✸✳ ❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ s♦♠❛ ✈❡♠♦s q✉❡A+D0 ={r ∈Q|r =a+b, para algum a∈
A e b ∈ D0}✳ ❙❡❥❛ a ∈ A✳ ❊♥tã♦ ♣❡❧❛ ♣❛rt❡ ✭■■■✮ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❈♦rt❡ ❞❡
❉❡❞❡❦✐♥❞ ❡①✐st❡ ✉♠ c∈A t❛❧ q✉❡ c < a✳ P♦rt❛♥t♦✱ a=c+ (a−c)∈A+D0✳
■ss♦ r❡s✉❧t❛ q✉❡ A ⊂ A+D0✳ ❆❣♦r❛✱ s❡ d ∈ A+D0✱ ❡♥tã♦ t❡♠♦s d =p+q
♦♥❞❡ p ∈ A ❡ q ∈ D0✳ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ D0 ✈❡♠♦s q✉❡ q > 0 ♦ q✉❡ ✐♠♣❧✐❝❛ p+q > p✳ P❡❧❛ ♣❛rt❡ ✭■■✮ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞ ❝♦♥❝❧✉í♠♦s q✉❡ d=p+q ∈A ✳ ■ss♦ r❡s✉❧t❛ q✉❡ A+D0 ⊂A ❡ ❛ss✐♠ t❡♠♦sA+D0 =A✳
✹✳ ❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ s♦♠❛ ❞❡ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞ ✈❡♠♦s q✉❡
A+ (−A) = {r∈Q|r=a+b, para algum a∈A e b ∈ −A}
❙❡❥❛x∈A+ (−A)✳ ❊♥tã♦x=p+q✱♦♥❞❡p∈A❡q∈ −A✳ ❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦
❞❡ −A t❡♠♦s q✉❡ −q < c ♣❛r❛ ❛❧❣✉♠ c∈Q−A✳ P❡❧❛ ♣❛rt❡ ✭■■✮ ❞♦ ❧❡♠❛ ✶✳✺
❝♦♥❝❧✉í♠♦s q✉❡ −q ∈Q−A✳ ■ss♦ r❡s✉❧t❛ q✉❡ −q < a,∀a ∈ A✱ ❡♠ ♣❛rt✐❝✉❧❛r
♣❛r❛ p∈ A✳ ❆ss✐♠✱ t❡♠♦s−q < p✱ ♦✉ s❡❥❛✱ x=p+q > 0✳ P♦rt❛♥t♦✱ x∈ D0✳
❉❡❞✉③✐♠♦s q✉❡ A+ (−A)∈ D0✳ ❆❣♦r❛✱ s❡ y ∈D0 t❡♠♦s y >0✳ ■ss♦ r❡s✉❧t❛
♣❡❧❛ ♣❛rt❡ ✭■✮ ❞♦ ❧❡♠❛ ✶✳✾ q✉❡ ❡①✐st❡♠ u∈A ❡ v ∈Q−A t❛✐s q✉❡ y=u−v
❝♦♠ v < e ♣❛r❛ ❛❧❣✉♠ e ∈ Q−A✳ ◆♦t❡ q✉❡ −(−v) < e ✐♠♣❧✐❝❛ −v ∈ −A✳
❆ss✐♠✱ y=u−v ∈A+ (−A)♦ q✉❡ ✐♠♣❧✐❝❛ D0 ∈A+ (−A)✳ P♦rt❛♥t♦✱ t❡♠♦s A+ (−A) =D0✳
❚❡♦r❡♠❛ ✷✳✷ ❙❡❥❛♠ A, B ∈ ℜ✳ ❊♥tã♦ sã♦ ✈á❧✐❞❛s ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿
✶✳ ❙❡ A+B =A+C✱❡♥tã♦ B =C✳ ✭▲❡✐ ❞♦ ❈❛♥❝❡❧❛♠❡♥t♦✮
✷✳ A=−(−A) ♣❛r❛ t♦❞♦ A∈ ℜ✳
✸✳ −(A+B) = (−A) + (−B) ♣❛r❛ t♦❞♦ A, B ∈ ℜ✳
❉❡♠♦♥str❛çã♦✿
✶✳ B = B +D0 = B + [A + (−A)] = (B +A) + (−A) = (A +B) + (−A) =
(A+C) + (−A) = (C+A) + (−A) = C+ [A+ (−A)] =C+D0 =C✳ P♦rt❛♥t♦✱
B =C✳
✷✳ A+ (−A) = (−A) +A = D0 = (−A) + [−(−A)] = [−(−A)] + (−A)✳ ■ss♦
r❡s✉❧t❛ ❞❡ ✭✶✮✱ q✉❡ A=−(−A)✳
✸✳ (A +B) + [−(A +B)] = D0 = D0 +D0 = [A + (−A)] + [B + (−B)] =
A+ [(−A) +B+ (−B)] =A+ [B+ (−A) + (−B)] = (A+B) + [(−A) + (−B)]✳
■ss♦ r❡s✉❧t❛ ❞❡ ✭✶✮✱ q✉❡ −(A+B) = (−A) + (−B)✳
❚❡♦r❡♠❛ ✷✳✸ ❙❡❥❛♠ A, B, C ∈ ℜ✳ ❊♥tã♦ sã♦ ✈á❧✐❞❛s ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿ ✶✳ ❖❝♦rr❡ ✉♠❛ ❞❛s s✐t✉❛çõ❡s ♦✉ A=B ♦✉ A > B ♦✉ A < B ✭▲❡✐ ❞❛❚r✐❝♦t♦♠✐❛✮
✷✳ AB =BA ✭▲❡✐ ❈♦♠✉t❛t✐✈❛✮