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EXISTÊNCIA E UNICIDADE DOS NÚMEROS REAIS VIA CORTES DE DEDEKIND

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❯♥✐✈❡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❡ ❞❡

❉❡❞❡❦✐♥❞✳

♣♦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♦ ❞❡

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❊①✐stê♥❝✐❛ ❡ ❯♥✐❝✐❞❛❞❡ ❞♦s

◆ú♠❡r♦s ❘❡❛✐s ✈✐❛ ❈♦rt❡ ❞❡

❉❡❞❡❦✐♥❞✳

♣♦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❇

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❆❣r❛❞❡❝✐♠❡♥t♦s

❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ♣r♦❢❡ss♦r ◆❛♣♦❧❡ó♥ ❈❛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✐❛ ♥❛❞❛✳

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❉❡❞✐❝❛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❡❧❧

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❘❡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♠♦❀

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❆❜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♠✳

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❙✉♠ár✐♦

✶ ❆❣r❛❞❡❝✐♠❡♥t♦s ✐✐✐

✷ ❈♦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 ✻✻

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■♥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✐❧♦❣✐❛ NZQ✱ ♠❛✐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 ❞❡

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❊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✐❧♦❣✐❛

ZNQ✱ 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✐❧♦❣✐❛ NZQ♦✉ZNQ✳

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❈❛♣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✉❡ r1, r+ 1 Q ❡ q✉❡ r1< r < r+ 1✱ ❞♦ q✉❛❧ t❡♠♦s q✉❡ r+ 1∈Q❡ r−1∈/ Q✳ P♦rt❛♥t♦✱ t❡♠♦sQ6=∅❡

D6=Q❀

■■✳ ❉❛❞♦ q✉❡xD❡ s✉♣♦♥❞♦ ❞❛❞♦ ✉♠yQt❛❧ q✉❡y > xt❡♠♦s q✉❡y > x > r✱

♣♦✐s✱ x∈D✳ P♦rt❛♥t♦✱ y∈D❀

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■■■✳ ❉❛❞♦ q✉❡xDt❡♠♦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)n1 +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} ♦♥❞❡ pQ✱ p >0✱ é ✉♠ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞✳

❉❡♠♦♥str❛çã♦✿ ❚❡♠♦s ❞♦✐s ❝❛s♦s ♣♦ssí✈❡✐s✿ ✭❛✮ ✵❁♣❁✶❀

✭❜✮ ♣❃✶❀

(12)

❊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♦ 2pM✱ ♣♦✐s✱ 2p >1> p >0 ❡ q✉❡ (2p)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✉❡ yx s❡❣✉❡✲s❡ q✉❡ yx >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 < xnnyxn−1 =xn(1ny x)≤x

n 1 y x

n

= (xy)n= x

2

n

< xn ♦ q✉❡ ✐♠♣❧✐❝❛ yn < xn ✱ ❧♦❣♦✱ ②❁①✳ 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 < xnp

nxn−1 < x ❡ 0 < s <

x n✳ ❋❛③❡♥❞♦ y = x s✱ ✈❡♠♦s q✉❡ y > 0✳ ❙❡❣✉❡✲s❡ s✉❝❡ss✐✈❛♠❡♥t❡✱ q✉❡

p < xnnsxn1 = xn(1ns x) ≤ x

n(1 s x)

n = (xs)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✳

(13)

✭❛✮ ◆♦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

e

nmpar✳ ❈❤❛♠❛♥❞♦ y= r

2 t❡♠♦s y > r ❡ y

n < p✱ ♣♦✐s✱ y <0✳ P♦rt❛♥t♦✱ t❡♠♦s q✉❡

y ∈ Dr ✐♠♣❧✐❝❛ 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=r1

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❡ ✉♠ yDr 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✳

▲❡♠❛ ✶✳✺ ❙❡❥❛ AQ ✉♠ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞✳

■✳ QA={xQ|x < a,aA}

■■✳ ❙❡❥❛ 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=aA✱ ♣♦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♦✱

QA ⊂ {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 {xQ|x < a,a A} ⊂QA✳ P♦rt❛♥t♦✱ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳

(14)

✭■■✮ P❡❧❛ ❤✐♣ót❡s❡✱ t❡♠♦s q✉❡yQ✱yx < a✱ ♣♦✐s✱xA✱ 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✳ ❉❛❞♦ aA✱ ❡♥tã♦ t❡♠♦s ❞♦✐s ❝❛s♦s✿

✶✳ ❊①✐st❡ ✉♠ aA t❛❧ q✉❡ a /B❀ ♦✉

✷✳ ❊①✐st❡ ✉♠ bB t❛❧ q✉❡ b /A✳

❈❛s♦ ✶✳ ❊♥tã♦ t❡♠♦s a A ❡ a QB✳ 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♦❀

✭■■✮ ❙❡❥❛ xB✳ ❙✉♣♦♥❤❛♠♦s ✉♠y Q t❛❧ q✉❡ yx✳ ▲♦❣♦✱ xX✱ ♣❛r❛ ❛❧❣✉♠ X ∈ A✳ ❖r❛✱ X é ✉♠ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞✱ s❡❣✉❡✲s❡ ❡♥tã♦ ♣❡❧❛ ♣❛rt❡ ✭■■✮ ❞❛

❞❡✜♥✐çã♦ q✉❡yX✱ ♦ q✉❡ ✐♠♣❧✐❝❛ yB✳ P♦rt❛♥t♦✱ ♣r♦✈❛♠♦s ❛ ♣❛rt❡ ✭■■✮ ❞❛

❞❡✜♥✐çã♦❀

✭■■■✮ ❙❡xB✱ ❡♥tã♦xX♣❛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✿

(15)

✭✐✮ ❖ ❝♦♥❥✉♥t♦ ❞❡✜♥✐❞♦ ♣♦rM ={r Q|r=a+b, para algum aA e bB}

é ✉♠ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞❀

✭✐✐✮ ❖ ❝♦♥❥✉♥t♦ ❞❡✜♥✐❞♦ ♣♦r M = {r Q| −r < c para algum c QA} é

✉♠ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞❀

✭✐✐✐✮ ❙✉♣♦♥❤❛ q✉❡ 0 ∈ Q−A ❡ 0 ∈ Q−B✳ ❖ ❝♦♥❥✉♥t♦ ❞❡✜♥✐❞♦ ♣♦r M = {r ∈

Q|r =ab, para algum aA e bB} é ✉♠ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞❀

✭✐✈✮ ❙✉♣♦♥❤❛ q✉❡ ❡①✐st❛ ✉♠ q QA 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❡♠ pQA ❡ qQB 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+ (sr)A✳ ❉❛í✱ s❡❣✉❡✲s❡ q✉❡ s= [a+ (sr)] +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✱ ♦♥❞❡cA ❡b B✳ ❆ss✐♠✱ s < r ❡✱ ♣♦rt❛♥t♦✱ sM✳

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✱

−(b1) M✳ ❆ss✐♠✱ ✜❝❛ ♣r♦✈❛❞♦ q✉❡M 6=✳ ❱❛♠♦s ♣r♦✈❛r q✉❡ a /M✱

✐st♦ é✱ −a QM✳ ❘❡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 ∈ QA 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♦✱ yM✳

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✭■■■✮ ❙❡ x M✱ ❡♥tã♦ ❡①✐st❡ ✉♠ c QA 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✉❡aA❡ ya B✳ ❆ss✐♠✱ ✈❡♠♦s q✉❡yM✳ 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 ❡ yM✳ P♦rt❛♥t♦✱ ❡①✐st❡ ✉♠ yM t❛❧ q✉❡ y < x✳

P❛rt❡ ✭✐✈✮

✭■✮ P❡❧❛ ❤✐♣ót❡s❡✱q >0❡q QA✱ ✐st♦ é✱ 0< q < a✱ ♣❛r❛ t♦❞♦a A✳ ❙❡rQ

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 = 2

q −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 < x1 < c ❝✉❥❛

❞❡s✐❣✉❛❧❞❛❞❡ r❡s✉❧t❛ ❡♠ 1

c < x✳ P♦❞❡♠♦s t♦♠❛r ✉♠y= x+1

c

2 ✳ ❆ss✐♠✱ ♣♦❞❡♠♦s

❝♦♥❝❧✉✐r q✉❡yQ ❡ 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❡❣✉❡✳

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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♦✮ ❙❡❥❛AN✉♠ s✉❜❝♦♥❥✉♥t♦ ♥ã♦✲

✈❛③✐♦✳ ❊♥tã♦ ❡①✐st❡ ✉♠ mA t❛❧ q✉❡ mx✱ ♣❛r❛ t♦❞♦ xA✳

▲❡♠❛ ✶✳✾ ❙❡❥❛ A ✉♠ ❈♦rt❡ ❞❡ ❉❡❞❡❦✐♥❞ ❡ y∈Q✳

✭■✮ ❙✉♣♦♥❞♦ q✉❡ y > 0✳ ❊♥tã♦ ❡①✐st❡♠ u A ❡ v QA t❛✐s q✉❡ y =uv✱ ❡ v < e✱ ♣❛r❛ ❛❧❣✉♠ eQA ❀

✭■■✮ ❙✉♣♦♥❤❛ q✉❡ y > 1 ❡ q✉❡ ❡①✐st❛ ✉♠ q ∈ QA t❛❧ q✉❡ q >0✳ ❊♥tã♦ ❡①✐st❡♠

r A❡ sQA 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✳ ❈♦♠♦

zw❡y sã♦ r❛❝✐♦♥❛✐s ♣♦s✐t✐✈♦s ❡①✐st❡ ✉♠nN t❛❧ q✉❡zw < ny✳ ❉❛ q✉❛❧

t✐r❛♠♦s ❡ r❡❡s❝r❡✈❡♠♦sz+n(y)< w✳ ❉♦ ▲❡♠❛ ✷✳✺ t✐r❛♠♦sz+n(y)QA✳

❱❛♠♦s ❞❡✜♥✐r ♦ ❝♦♥❥✉♥t♦ M t❛❧ q✉❡✿

M ={pN|z+p(y)QA}.

❆♥t❡r✐♦r♠❡♥t❡✱ ❝♦♠♦ ❡①✐st❡ n ∈ N t❛❧ q✉❡ z+n(−y) ∈ QA s❡❣✉❡✲s❡ q✉❡ M 6= ✳ ❖r❛✱ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ❇♦❛ ❖r❞❡♥❛çã♦✱❡①✐st❡ ✉♠ m N t❛❧ q✉❡

z+m(y)QA ❡z+ (m1)(y)A✳ ❉✐ss♦ r❡s✉❧t❛ ♣❡❧♦ ▲❡♠❛ ✷✳✺ ✭■■✮

q✉❡✱ ❡①✐st❡ ✉♠e∈QAt❛❧ q✉❡z+m(−y)< e✳ ❋❛ç❛♠♦su=z+(m−1)(−y)

❡v =z+m(y)✳ ❉♦ q✉❛❧ t✐r❛♠♦sy=uv✱ ❝♦♠v < e✱ ♣❛r❛ ❛❧❣✉♠eQA✳

✭■■✮ ❆ ❞❡♠♦♥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✐❣✉❛❧❞❛❞❡ (y1)v > uv ✭❝✉❥❛ ✐♠✲

♣❧✐❝❛çã♦ ♥♦s ❞á y > u

v✮✳ P❛r❛ q✉❡ ✐ss♦ ♦❝♦rr❛✱ é ♥❡❝❡ssár✐♦ q✉❡ ❡❧❛ ❛❞✈❡♥❤❛ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡(y1)v >(y1)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♦ ✜❝❛

(18)

r❡s♦❧✈✐❞❛✳ ❱❡❥❛♠♦s✿ ❈♦♠♦y >1❡q >0t❡♠♦s(y1)q2 >0✳ ▲♦❣♦✱ ♣❡❧❛ ♣❛rt❡

✭■✮ ❞❡ss❡ ❧❡♠❛ t❡♠♦s q✉❡ ❡①✐st❡♠ u∈A❡v ∈Q−A t❛✐s q✉❡(y−1)q2 =u−v

❝♦♠ v < g✱ ♣❛r❛ ❛❧❣✉♠ g ∈ QA✳ ❙❡ ❡st❡ v ❢♦r t❛❧ q✉❡ v > q2✱ ❡♥tã♦ t❡r❡✲

♠♦s (y1)v > (y1) q2 = uv✱ ♦✉ s❡❥❛✱ (y1)v > uv ♦ q✉❡ ✐♠♣❧✐❝❛

s✉❝❡ss✐✈❛♠❡♥t❡ ❡♠ yv−v > u−v ❡ y > u

v✱ ♣♦✐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❡♠rA❡sQAt❛✐s q✉❡y > r

s ❝♦♠s >0❡s < g✱♣❛r❛ ❛❧❣✉♠g ∈Q−A✳ ❆❣♦r❛✱ ✈❛♠♦s ❝♦♥str✉✐r ♦ r❛❝✐♦❝í♥✐♦ q✉❛♥❞♦ v 2q ✳ ❆ ✐❞❡✐❛ ❝♦♥s✐st❡ ❡♠

❝♦♥str✉✐r ♦✉tr♦s ❡❧❡♠❡♥t♦s r ∈ A ❡ s ∈ Q− A t❛✐s q✉❡ ❛✐♥❞❛ t❡♥❤❛♠♦s

(y1)q2 =uv =rs ❞❡✈❡♠♦s ❢❛③❡r q✉❡ ♦❝♦rr❛(y1)s >(y1)q2 =rs

❡ q✉❡ s = 34q✳ ❆❣♦r❛✱ ❜❛st❛ ♠♦str❛r♠♦s ❝♦♠ ✐ss♦ q✉❡ r A ❡ s QA ✳

❈♦♠ ❡❢❡✐t♦✱ é ✐♠❡❞✐❛t♦ q✉❡ ♣♦r s = 34 q < 12q < q r❡s✉❧t❛ s ∈ Q−A✱ ❡ q✉❡

♣♦r r = u+ (sv) > u ✭♣♦✐s✱ s > q

2 ≥ v✮ r❡s✉❧t❛ r ∈ A✳ ❆ss✐♠✱ ❛ s❡❣✉♥❞❛

♣❛rt❡ ❞♦ ❧❡♠❛ ✜❝❛ ♣r♦✈❛❞♦✳ ❱❡❥❛♠♦s✿ ❋❛ç❛♠♦s r=u+ (sv) ❡ s❡❥❛ s= 34q✳

❆ ♣❛rt✐r ❞✐ss♦ t❡♠♦s r−s =u−v = (y−1)q2 <(y−1)s ❡✱ ♣♦rt❛♥t♦✱ r < sy

♦ q✉❡ ✐♠♣❧✐❝❛ y > r

s✳ ❆❣♦r❛✱ ❝♦♠♦ s =

3

4q < q ❡ q ∈ Q−A ❝♦♥❝❧✉í♠♦s q✉❡

s QA ❡ q✉❡✱ ♣♦r tr❛♥s✐t✐✈✐❞❛❞❡✱ s < q = g QA✳ ❖r❛✱ ✈❡♠♦s q✉❡ r =u+ (s−v) > u✱ ♣♦✐s✱ s−v > 0✳ ▲♦❣♦✱ r ∈ A✳ P♦rt❛♥t♦✱ ♣r♦✈❛♠♦s q✉❡

❡①✐st❡♠ rA ❡s QAt❛✐s q✉❡ y > r

s ❝♦♠ s < g✱ ♣❛r❛ ❛❧❣✉♠ g ∈Q−A ✳

(19)

❈❛♣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 aA e bB}.

(20)

❉❡✜♥✐çã♦ ✷✳✹ ❉❛❞♦ A ∈ ℜ ✱ ❞❡✜♥✐♠♦s ♦♣♦st♦ ❞❡ A✱ ❞❡♥♦t❛❞♦ ♣♦r A ♦ ♥ú♠❡r♦

r❡❛❧

−A ={r ∈Q| −r < c para algum c∈QA}.

❆ ❞❡✜♥✐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❡♠♦sADr ❡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❡ qQA t❛❧ q✉❡ r < q✱ t❡♠♦s ♣❡❧♦

❧❡♠❛ ✷✳✼✱ q✉❡ r < qa,aA✳ ❆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❡ ❛ rQA ✳

✭✐✐✐✮ ❙✉♣♦♥❤❛ 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 / QA✱ ❡ ♣♦rt❛♥t♦ 0 / QA✱ ♦ q✉❡ ✐♠♣❧✐❝❛ 0 ∈ −/ A✳ P❡❧❛ ♣❛rt❡ ✭■■✮

❞❡st❡ ❧❡♠❛ ❝♦♥❝❧✉í♠♦s q✉❡ −AD0 ♦ 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 ✳

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✭■■■✮ A·B =[A·(B)]✱ s❡ AD0 ❡ 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 ={rQ|r >0 e 1

r < c para algum c∈Q−A}✱ s❡ A < D0✳ ✭■■✮ A−1 =(A)−1✱ s❡ A > D

0✳

❈♦♠ ❛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 bB}={rQ|r =b+a para algum bB e aA}=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❡ ✉♠ cA t❛❧ q✉❡ c < a✳ P♦rt❛♥t♦✱ a=c+ (ac)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✳

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✹✳ ❯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∈QA✳ P❡❧❛ ♣❛rt❡ ✭■■✮ ❞♦ ❧❡♠❛ ✷✳✺

❝♦♥❝❧✉í♠♦s q✉❡ −q QA✳ ■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❡♠ uA ❡ v QA t❛✐s q✉❡ y=uv

❝♦♠ 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✐✈❛✮

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✸✳ A(BC) = (AB)C ✭▲❡✐ ❆ss♦❝✐❛t✐✈❛ ♣❛r❛ ❛ ▼✉❧t✐♣❧✐❝❛çã♦✮

✹✳ AD1 =A ✭▲❡✐ ❞❛ ■❞❡♥t✐❞❛❞❡ ▼✉❧t✐♣❧✐❝❛t✐✈❛✮

✺✳ ❙❡ A 6=D0✱ ❡♥tã♦ AA−1 =D1 ✭▲❡✐ ❞♦ ■♥✈❡rs♦ ▼✉❧t✐♣❧✐❝❛t✐✈♦✮

✻✳ A(B+C) =AB +AC ✭▲❡✐ ❞❛ ❉✐str✐❜✉t✐✈❛✮

❉❡♠♦♥str❛çã♦✿

✶✳ ❊ss❛ ♣❛rt❡ s❡❣✉❡✲s❡ ✐♠❡❞✐❛t❛♠❡♥t❡ ❞♦ ▲❡♠❛ ✷✳✻ ❡ ❞❛ ❞❡✜♥✐çã♦ s♦❜r❡ ❛ r❡❧❛çã♦ ❞❡ ♦r❞❡♠✳

✷✳ P❡❧❛ ♣❛rt❡ ✭✶✮ ❞❡st❡ t❡♦r❡♠❛✱ s❛❜❡♠♦s q✉❡ ♦✉ A D0 ♦✉ A < D0 ♦ ♠❡s♠♦

♦❝♦rr❡♥❞♦ ❝♦♠ B✳ ■ss♦ r❡s✉❧t❛ q✉❛tr♦ ❝❛s♦s✿

• Pr✐♠❡✐r♦✱ s✉♣♦♥❤❛ q✉❡ A≥D0 ❡ B ≥D0✳ ❊♥tã♦

AB ={r ∈Q|r=ab, para algum a∈A e b∈B}

={r Q|r =ba, para algum aA e bB}=BA.

• ❙❡❣✉♥❞♦✱ s✉♣♦♥❞♦ A D0 ❡ B < D0 ✈❡♠♦s ♣❡❧❛ ♣❛rt❡ ✭■■❧✮ ❞♦ ▲❡♠❛

✸✳✶ q✉❡ −B ≥ D0 ✳ ❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ r❡❛✐s t❡♠♦s

AB =[A·(B)] = [(B)·A] =(B)·A=BA✱ ♣♦✐s✱ (A) =A✳

• ❚❡r❝❡✐r♦✱ ♦ ❝❛s♦A < D0 ❡B ≥D0 é ❛♥á❧♦❣♦ ❛♦ s❡❣✉♥❞♦ ❝❛s♦✳

• ◗✉❛rt♦✱ s✉♣♦♥❞♦ A < D0 ❡ B < D0 ✈❡♠♦s ♣❡❧❛ ♣❛rt❡ ✭■■■✮ ❞♦ ▲❡♠❛

✸✳✶ q✉❡ −A ≥ D0 ❡ −B ≥ D0✳ ❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ t❡r❡♠♦s AB =

(A)(B) = (B)(A) =BA.

✸✳ ❚❡♠♦s ♦✐t♦ ❝❛s♦s✳

• Pr✐♠❡✐r♦✱ s✉♣♦♥❤❛ A D0✱ B ≥ D0✱ ❡ C ≥ D0✳ ❙❡ A ≥ D0✱ B ≥

D0 ❡ C ≥ D0 ❡♥tã♦ é ❢á❝✐❧ ❝♦♥❝❧✉✐r✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦✱

q✉❡ AB D0 ❡ BC ≥ D0✳ ❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ✈❡✲

♠♦s q✉❡ A(BC) = {r Q|r = ad para algum a A e d BC e se, e somente se, para algum b∈B e c∈C tais que d=

bc} = {r Q|r = a(bc) para algum a A, b B e c C} = {r

Q|r = (ab)c para algum a A, b B e c C} = {r Q|r = (ab)c para algum (ab)∈AB e c∈C}= (AB)C.

• ❙❡❣✉♥❞♦✱ s✉♣♦♥❤❛ q✉❡ AD0✱ B < D0✱ ❡C ≥D0 ✳ ❊♥tã♦ é ❢á❝✐❧✱ t❛♠✲

❜é♠✱ ❝♦♥❝❧✉✐r q✉❡ −B ≥D0✱ AB < D0 ❡ BC < D0✳ ❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦

❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ✈❡♠♦s q✉❡ A(BC) = [A(BC)] = −{A[(B)C]} =

−{[A(B)]C}=[A(B)]C= (AB)C✳

Referências

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