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Question Number 81157 by ~blr237~ last updated on 09/Feb/20
![Let n≥2 , for x∈[0,1] : let consider A(x)={ u∈R_+ ^∗ \ x<u^n } 1)Prove that if a,b∈[0,1] a≤b ⇔A(a)⊆A(b) 2)Deduce x=[infA(x) ]^n](Q81157.png)
$${Let}\:{n}\geqslant\mathrm{2}\:,\:{for}\:\:{x}\in\left[\mathrm{0},\mathrm{1}\right]\:\::\:\:\:{let}\:\:{consider}\:\:{A}\left({x}\right)=\left\{\:{u}\in\mathbb{R}_{+} ^{\ast} \:\backslash\:\:\:{x}<{u}^{{n}} \right\}\: \\ $$ $$\left.\mathrm{1}\right){Prove}\:\:{that}\:{if}\:\:\:{a},{b}\in\left[\mathrm{0},\mathrm{1}\right]\:\:\:\:\:\:\:\:\:\:{a}\leqslant{b}\:\Leftrightarrow{A}\left({a}\right)\subseteq{A}\left({b}\right)\:\:\: \\ $$ $$\left.\mathrm{2}\right){Deduce}\:\:\:\:\:{x}=\left[{infA}\left({x}\right)\:\right]^{{n}} \:\: \\ $$