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Question Number 192297 by Mastermind last updated on 14/May/23

Answered by witcher3 last updated on 14/May/23

$$\mathrm{let}\mathrm{a}=\mathrm{supA},\mathrm{b}=\mathrm{SupB}\Rightarrow \mathrm{\forall }(\mathrm{x}\in \mathrm{Aety}\in \mathrm{B})$$$$\mathrm{x}+\mathrm{y}⩽\mathrm{a}+\mathrm{b}\Rightarrow \sup (\mathrm{A}+\mathrm{B})⩽\mathrm{a}+\mathrm{b}$$$$\mathrm{let}\mathrm{M}=\sup (\mathrm{A}+\mathrm{B})$$$$\mathrm{\forall }ϵ>0\mathrm{\exists }\mathrm{x}\in \mathrm{A},\mathrm{y}\in \mathrm{B}\mathrm{such}\mathrm{a}-ϵ<\mathrm{x}⩽\mathrm{a}$$$$\mathrm{b}-ϵ<\mathrm{y}⩽\mathrm{b}$$$$\Rightarrow \mathrm{a}+\mathrm{b}-2ϵ<\mathrm{x}+\mathrm{y}\in \mathrm{A}+\mathrm{B}⩽\mathrm{a}+\mathrm{b}$$$$\Rightarrow \mathrm{\forall }ϵ>0\mathrm{\exists }\mathrm{S}=\mathrm{x}+\mathrm{y}\in \mathrm{A}+\mathrm{B}$$$$\mathrm{such}\mathrm{a}+\mathrm{b}-ϵ<\mathrm{x}+\mathrm{y}$$$$\Rightarrow \sup (\mathrm{A}+\mathrm{B})⩾\mathrm{a}+\mathrm{b}=\sup \left(\mathrm{A}\right)+\sup \left(\mathrm{B}\right)$$$$\Rightarrow \sup (\mathrm{A}+\mathrm{B})=\sup \left(\mathrm{A}\right)+\sup \left(\mathrm{B}\right)$$$$$$

Commented by Mastermind last updated on 14/May/23

$$\mathrm{I}\mathrm{do}\mathrm{really}\mathrm{appreciate}$$$$$$$$\mathrm{Thank}\mathrm{you}\mathrm{man}$$

Answered by witcher3 last updated on 14/May/23

$$\left(2\right),{\mathrm{a}}_{\mathrm{n}}=\arctan \left(\mathrm{n}\right),{\mathrm{b}}_{\mathrm{n}}=\arctan (-\mathrm{n})$$$${\mathrm{a}}_{\mathrm{n}}+{\mathrm{b}}_{\mathrm{n}}=0$$$${\mathrm{supa}}_{\mathrm{n}}+{\mathrm{supb}}_{\mathrm{n}}=\pi$$$$$$

Commented by Mastermind last updated on 14/May/23

$$\mathrm{Thank}\mathrm{you}\mathrm{my}\mathrm{BOSS}$$$$\mathrm{is}\mathrm{that}\mathrm{all}?$$

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