❚é❝♥✐❝❛s ♣❛r❛ ❉❡♠♦♥str❛çã♦ ❞❡
❚❡♦r❡♠❛s ✖ P❛rt❡ ■
▼❛t❡♠át✐❝❛ ❉✐s❝r❡t❛ ■
❘♦❞r✐❣♦ ❘✐❜❡✐r♦
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❆♣❧✐❝❛❞❛s ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❋❡❞❡r❛❧ ❞❡ ❖✉r♦ Pr❡t♦
▼♦t✐✈❛çã♦ ✖ ✭■✮
P♦rq✉ê ❛♣r❡♥❞❡r ❞❡♠♦♥str❛çã♦❄
❆❥✉❞❛ ❛ ❞❡s❡♥✈♦❧✈❡r ❛ ❤❛❜✐❧✐❞❛❞❡ ❞❡ ♥ã♦ s♦♠❡♥t❡ ❝r✐❛r ❞❡♠♦♥str❛çõ❡s✱ ♠❛s s♦❜r❡t✉❞♦ ❡♥t❡♥❞ê✲❧❛s✳ ❖ ❝♦♥❤❡❝✐♠❡♥t♦ ❡♠ ❝r✐❛r ❞❡♠♦♥str❛çõ❡s
♠❛♥✉❛❧♠❡♥t❡ é ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ❛
▼♦t✐✈❛çã♦ ✖ ✭■■✮
❚é❝♥✐❝❛s ♣❛r❛ ❞❡♠♦♥str❛çã♦
P❛r❛ ❞❡♠♦♥str❛r t❡♦r❡♠❛s✱ ❞❡✈❡♠♦s ❡①♣r❡ss❛r ♦ q✉❡ ❞❡s❡❥❛♠♦s ♣r♦✈❛r ✉s❛♥❞♦ ❧ó❣✐❝❛ ❞❡
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭■✮
❆❧❣✉♠❛s ❉❡✜♥✐çõ❡s✳✳✳
❯♠ t❡♦r❡♠❛ é ❢♦r♠❛❞♦ ♣❡❧♦s s❡❣✉✐♥t❡s ❝♦♠♣♦♥❡♥t❡s✿ ❍✐♣ót❡s❡s ❡ ❈♦♥❝❧✉sã♦✳
❚❡♦r❡♠❛s sã♦ s❡♥t❡♥ç❛s q✉❡ ❞❡s❝r❡✈❡♠ ❛❧❣✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ♠❛t❡♠át✐❝❛✳
❚❛✐s s❡♥t❡♥ç❛s ♣♦❞❡♠ s❡r ✈❡r❞❛❞❡✐r❛s ♦✉ ❢❛❧s❛s✳ P❛r❛ ♠♦str❛r q✉❡ ✉♠ t❡♦r❡♠❛ é ✈❡r❞❛❞❡✐r♦ ❞❡✈❡♠♦s ♣r♦✈á✲❧♦✳
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭■■✮
❚❡♦r❡♠❛
❙❡❥❛♠ ❛,❜ ∈ R. ❙❡ ✵ < ❛ < ❜ ❡♥tã♦ ❛✷ < ❜✷✳
❊st❡ ❡①❡♠♣❧♦ ♠♦str❛ ✉♠ s✐♠♣❧❡s t❡♦r❡♠❛ ❡①♣r❡ss♦ ❡♠ t❡r♠♦s ❞❛s ✈❛r✐á✈❡✐s ❛,❜✳
P❛r❛ ♣r♦✈á✲❧♦ ❞❡✈❡♠♦s ♠♦str❛r q✉❡ s❡ ✵ < ❛ < ❜
❡♥tã♦ ❛✷ < ❜✷✱ ♣❛r❛ q✉❛✐sq✉❡r ✈❛❧♦r❡s ❞❡ ❛,❜✳
❊st❡ t❡♦r❡♠❛ ♣♦ss✉✐ ❝♦♠♦ ❤✐♣ót❡s❡ ❛,❜ ∈ R
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭■■■✮
❈♦♥❝❡✐t♦ ❞❡ ■♥stâ♥❝✐❛
❈♦♥s✐❞❡r❡ ❛✐♥❞❛ ♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r✳✳✳
❙❡❥❛♠ ❛,❜ ∈ R. ❙❡ ✵< ❛< ❜ ❡♥tã♦ ❛✷ < ❜✷✳
❯♠❛ ✐♥stâ♥❝✐❛s ❞❡ ✉♠ t❡♦r❡♠❛ é ♦❜t✐❞❛
s✉❜st✐t✉✐♥❞♦ ✈❛r✐á✈❡✐s ♣♦r ✈❛❧♦r❡s ♥♦ ❡♥✉♥❝✐❛❞♦ ❞♦ t❡♦r❡♠❛✳
❊①✿ ❙❡ ✵ < ✺ < ✼ ❡♥tã♦ ✷✺< ✹✾✱ ♣❛r❛
❛ = ✺,❜ = ✼✳
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭■❱✮
❊①❡r❝í❝✐♦
❙✉♣♦♥❤❛ ①,② ∈ R t❛✐s q✉❡ ① > ✸ ❡ ② < ✷✳ ❊♥tã♦
①✷ −✷② > ✺✳
◗✉❛✐s sã♦ ❛s ❤✐♣ót❡s❡s ❡ ❛ ❝♦♥❝❧✉sã♦ ❞❡st❡ t❡♦r❡♠❛❄
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❱✮
❚❡♦r❡♠❛
❙✉♣♦♥❤❛ ①,② ∈ R t❛✐s q✉❡ ① > ✸✳ ❊♥tã♦
①✷ −✷② > ✺✳
❈♦♥tr❛✲❡①❡♠♣❧♦s sã♦ ✉t✐❧✐③❛❞♦s ♣❛r❛ ♠♦str❛r q✉❡ ✉♠ t❡♦r❡♠❛ é ❢❛❧s♦✳
❯♠ ❝♦♥tr❛✲❡①❡♠♣❧♦ ♣❛r❛ ♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r é ① = ✹,② = ✻✱ ♣♦✐s ❝♦♠ ✐ss♦ t❡♠♦s
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❱■✮
P❛r❛ ♥ã♦ ❡sq✉❡❝❡r✳✳✳
P❛r❛ ❛✜r♠❛r q✉❡ ✉♠ t❡♦r❡♠❛ é ✈❡r❞❛❞❡✐r♦✱ ❞❡✈❡♠♦s ❝♦♥str✉✐r ✉♠❛ ♣r♦✈❛✳ ❙♦♠❡♥t❡ ❛ss✐♠ ❣❛r❛♥t✐♠♦s q✉❡ ❡st❡ é ✈❡r❞❛❞❡✐r♦ ♣❛r❛ t♦❞❛s ❛s s✉❛s ✐♥stâ♥❝✐❛s✳
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❱■■✮
❉❡♠♦♥str❛♥❞♦ ✉♠ t❡♦r❡♠❛✳✳✳
P❛r❛ ❞❡♠♦♥str❛r ✉♠ t❡♦r❡♠❛s✱ t❡♠♦s ✉♠❛ ❡s♣é❝✐❡ ❞❡ ✏r❡❝❡✐t❛ ❞❡ ❜♦❧♦✑✿
❊①♣r❡ss❡ ♦ t❡♦r❡♠❛ ❛ s❡r ❞❡♠♦♥str❛❞♦ ❝♦♠♦ ✉♠❛ ❢ór♠✉❧❛ ❞❛ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s✳
❉❡♠♦♥str❡ ♦ t❡♦r❡♠❛✱ ❢❛③❡♥❞♦ t♦❞❛s ❛s ❞❡❞✉çõ❡s ♣❛ss♦ ❛ ♣❛ss♦✳ ❆s ❞❡❞✉çõ❡s sã♦ s✐♠✐❧❛r❡s ❛♦ q✉❡ ❥á ❡st✉❞❛♠♦s ❞❡ ❞❡❞✉çã♦ ♥❛t✉r❛❧✳
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❱■■■✮
❉❡♠♦♥str❛♥❞♦ ✉♠ t❡♦r❡♠❛✳✳✳
◆❡st❛ ❞✐s❝✐♣❧✐♥❛ s❡❣✉✐r❡♠♦s ❛ ❛❜♦r❞❛❣❡♠ ❞♦ ❧✐✈r♦ ✏❍♦✇ t♦ Pr♦✈❡ ✐t✑ q✉❡ ❞✐✈✐❞❡ ❛
❞❡♠♦♥str❛çã♦ ❞❡ ✉♠ t❡♦r❡♠❛ ❡♠ ❞✉❛s ♣❛rt❡s✳
❘❛s❝✉♥❤♦✿ ◆♦ r❛s❝✉♥❤♦ sã♦ ❢❡✐t❛s t♦❞❛s ❛s ❞❡❞✉çõ❡s ♥❡❝❡ssár✐❛s ♣❛r❛ ❛ ❝♦♥❝❧✉sã♦ ❞❡ q✉❡ ♦ t❡♦r❡♠❛ é ✈❡r❞❛❞❡✐r♦✳
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭■❳✮
❘❛s❝✉♥❤♦
❖ r❛s❝✉♥❤♦ é ❝♦♠♣♦st♦ ❞❡ ❞✉❛s ♣❛rt❡s✿
❍✐♣ót❡s❡s✿ ❈♦♥❥✉♥t♦ ❞❡ ❢ór♠✉❧❛s q✉❡ ❢♦r❛♠ ❞❛❞❛s ❝♦♠♦ ❤✐♣ót❡s❡s ♣❡❧♦ ❡♥✉♥❝✐❛❞♦ ❞♦
t❡♦r❡♠❛ ❡ s✉♣♦s✐çõ❡s ❢❡✐t❛s ❞✉r❛♥t❡ ♦ ♣r♦❝❡ss♦ ❞❡ ❞❡♠♦♥str❛çã♦✳
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳✮
❘❡❝❛♣✐t✉❧❛♥❞♦✳✳✳
❖s ♣❛ss♦s ♣❛r❛ ❞❡♠♦♥str❛r ✉♠ t❡♦r❡♠❛ sã♦✿ ❊①♣r❡ss❡ ♦ ❡♥✉♥❝✐❛❞♦ ❞♦ t❡♦r❡♠❛ ❝♦♠♦ ✉♠❛ ❢ór♠✉❧❛ ❞❛ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s✳
❈♦♥str✉❛ ✉♠ r❛s❝✉♥❤♦ ❡ ❞❡♠♦♥str❡ ❛ ❝♦♥❝❧✉sã♦ ❞♦ t❡♦r❡♠❛ ❛ ♣❛rt✐r ❞❡ s✉❛s ❤✐♣ót❡s❡s✳
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳■✮
Pr✐♠❡✐r♦ ❊①❡♠♣❧♦✳✳✳
❚❡♦r❡♠❛ ✶✿ ❙✉♣♦♥❤❛ ❛,❜ ∈ R✳ ❙❡ ✵ < ❛ < ❜
❡♥tã♦ ❛✷ < ❜✷✳
❘❡♣r❡s❡♥t❛♥❞♦ ♦ t❡♦r❡♠❛ ❝♦♠♦ ✉♠❛ ❢ór♠✉❧❛ ❧ó❣✐❝❛
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳■■✮
❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✶
❍✐♣ót❡s❡s Pr♦✈❛r
❛,❜ ∈ R ✵ < ❛ < ❜ →❛✷ < ❜✷
P♦❞❡♠♦s ♣❡r❝❡❜❡r q✉❡ ❛ ❢ór♠✉❧❛ ♣♦ss✉✐ ❛ ❢♦r♠❛ P →◗✳
P❛r❛ ♣r♦✈❛r P →◗ ♣♦❞❡♠♦s ✉s❛r (→ ■)✱ ✐st♦
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳■■■✮
❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✷
❆♣ós s✉♣♦r ♦ ❧❛❞♦ ❡sq✉❡r❞♦ ❞❛ ✐♠♣❧✐❝❛çã♦ t❡♠♦s ♦ s❡❣✉✐♥t❡ r❛s❝✉♥❤♦✿
❍✐♣ót❡s❡s Pr♦✈❛r ❛,❜ ∈ R ❛✷ < ❜✷
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳■❱✮
❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✸
❈♦♠♦ ❛,❜ ∈ R ❡ ✵ < ❛ < ❜ t❡♠♦s q✉❡ ❛ < ❜✳
❆❧é♠ ❞✐ss♦✱ ♣♦❞❡♠♦s ♠✉❧t✐♣❧✐❝❛r ❛♠❜♦s ♦s ❧❛❞♦s ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛ < ❜ ♣♦r ❛ ♦❜t❡♥❞♦ ♦ s❡❣✉✐♥t❡
r❛s❝✉♥❤♦✿
❍✐♣ót❡s❡s Pr♦✈❛r ❛,❜ ∈ R ❛✷ < ❜✷
✵ < ❛< ❜
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❱✮
❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✹
▼✉❧t✐♣❧✐❝❛♥❞♦ ❛♠❜♦s ♦s ❧❛❞♦s ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛ < ❜ ♣♦r ❜ ♦❜t❡♥❞♦ ♦ s❡❣✉✐♥t❡ r❛s❝✉♥❤♦✿
❍✐♣ót❡s❡s Pr♦✈❛r ❛,❜ ∈ R ❛✷ < ❜✷
✵ < ❛< ❜
❛✷ < ❛❜
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❱■✮
❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✺
❖❜s❡r✈❡ q✉❡ s❡ ❛✷ < ❛❜ ❡ ❛❜ < ❜✷ t❡♠♦s
❛✷ < ❛❜ < ❜✷✳
❍✐♣ót❡s❡s Pr♦✈❛r ❛,❜ ∈ R ❛✷ < ❜✷
✵ < ❛ < ❜
❛✷ < ❛❜
❛❜ < ❜✷
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❱■■✮
❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✻
❋✐♥❛❧♠❡♥t❡ ❝♦♠♦ t❡♠♦s q✉❡ ❛✷ < ❛❜ < ❜✷✱
♣♦❞❡♠♦s ❝♦♥❝❧✉✐r ❛✷ < ❜✷✳
❍✐♣ót❡s❡s Pr♦✈❛r ❛,❜ ∈ R ❛✷ < ❜✷
✵ < ❛ < ❜
❛✷ < ❛❜
❛❜ < ❜✷
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❱■■■✮
❊ ❡♥tã♦✳✳✳
❈♦♠♦ ♦❜t❡♠♦s ❛ ❞❡❞✉çã♦ ❞❛ ❝♦♥❝❧✉sã♦ ❛✷ < ❜✷
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳■❳✮
❚❡①t♦ ❞❛ Pr♦✈❛
❚❡♦r❡♠❛ ✶✿ ❙✉♣♦♥❤❛ ❛,❜ ∈ R✳ ❙❡ ✵ < ❛ < ❜
❡♥tã♦ ❛✷ < ❜✷✳
Pr♦✈❛✿ ❙✉♣♦♥❤❛ ✵ < ❛ < ❜✳ ▼✉❧t✐♣❧✐❝❛♥❞♦ ❛
❞❡s✐❣✉❛❧❞❛❞❡ ❛< ❜ ♣♦r ❛ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡
❛✷ < ❛❜ ❡ s✐♠✐❧❛r♠❡♥t❡ ♠✉❧t✐♣❧✐❝❛♥❞♦ ♣♦r ❜ ♦❜t❡♠♦s
❛❜ < ❜✷✳ ▲♦❣♦✱ t❡♠♦s q✉❡ ❛✷ < ❛❜ < ❜✷ ❡ ♣♦rt❛♥t♦
❛✷ < ❜✷ ❝♦♠♦ r❡q✉❡r✐❞♦✳ ❆ss✐♠✱ s❡ ✵ < ❛ < ❜ ❡♥tã♦
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳✮
❊str❛té❣✐❛s ❞❡ Pr♦✈❛
❆ ♣❛rt✐r ❞♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r ♣♦❞❡♠♦s ❞❡❞✉③✐r ♥♦ss❛ ♣r✐♠❡✐r❛ ❡str❛té❣✐❛ ❞❡ ♣r♦✈❛✿ Pr♦✈❛r ❛✜r♠❛t✐✈❛s ❞❛ ❢♦r♠❛ P →◗✳
❆s ❡str❛té❣✐❛s ❞❡ ♣r♦✈❛ s❡rã♦ ♦r❣❛♥✐③❛❞❛s ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿
Pr✐♠❡✐r❛♠❡♥t❡ s❡rá ❛♣r❡s❡♥t❛❞♦ ♦ ❢♦r♠❛t♦ ❞♦ r❛s❝✉♥❤♦ ❛♥t❡s ❞❛ ✉t✐❧✐③❛çã♦ ❞❛ ❡str❛té❣✐❛✳ ❊♠ s❡❣✉✐❞❛ s❡rá ❛♣r❡s❡♥t❛❞♦ ♦ r❛s❝✉♥❤♦ ❛♣ós ❛ ✉t✐❧✐③❛çã♦ ❞❛ ❡str❛té❣✐❛✳
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳■✮
❊str❛té❣✐❛ ❞❡ Pr♦✈❛ ✶
P❛r❛ ♣r♦✈❛r ✉♠❛ ❛✜r♠❛t✐✈❛ P →◗✿
❘❛s❝✉♥❤♦ ❛♥t❡s ❞❡ ✉s❛r ❛ ❡str❛té❣✐❛✿ ❍✐♣ót❡s❡s Pr♦✈❛r
... P →◗
❘❛s❝✉♥❤♦ ❞❡♣♦✐s ❞❡ ✉s❛r ❛ ❡str❛té❣✐❛✿ ❍✐♣ót❡s❡s Pr♦✈❛r
... ◗
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳■■✮
❊str❛té❣✐❛ ❞❡ Pr♦✈❛ ✶
P❛r❛ ♣r♦✈❛r ✉♠❛ ❛✜r♠❛t✐✈❛ P →◗✿
❚❡①t♦ ✜♥❛❧ ❞❛ ♣r♦✈❛✿
❙✉♣♦♥❤❛ P✳
❬❚❡①t♦ ❞❛ ♣r♦✈❛ ❞❡ ◗❪✳
❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳■■■✮
❊①❡r❝í❝✐♦s
❚❡♦r❡♠❛ ✶✿ ❙✉♣♦♥❤❛ q✉❡ ❛,❜ ∈ R✳ ❙❡ ❛ < ❜
❡♥tã♦ ❛+❜
✷ ✳
❚❡♦r❡♠❛ ✷✿ ❙✉♣♦♥❤❛ q✉❡ ❆\❇ ⊆❈ ∩❉ ❡
