Suppose $X \in (0,1)$, and that $X^n < (1-e^{n\delta})$ for $\delta>0$ and $n \in \mathbb{N}$, then, I am trying to see if
$$ X < (1-e^{n\delta})^{1/n} $$
is true? In other words, is the $n$th root a monotone function here?
Suppose $X \in (0,1)$, and that $X^n < (1-e^{n\delta})$ for $\delta>0$ and $n \in \mathbb{N}$, is it true $X < (1-e^{n\delta})^{1/n}$?
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real-analysis
monotone-functions
1 Answers
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The $n$-th root function is strictly increasing on the set $E$ of non-negative real numbers (note that said function may not exist elsewhere, for we don't know the parity of $n$).
To see why, say $a,b \in E$ with $a