Evaluation of limit of the sequence

Question

Let $x_n=Sup\lbrace x^n: 0<x<1 \rbrace$. Then what will be the limit of $x_n$ as $n\to\infty$ ?

I think the answer will be either $0$ or $1$. But I am not getting the rigorous argument about the answer. I know that we have to use the fact : $\lim_{n\to\infty}x^n=0$ provided $0\leq x<1$.

Answer 1

Note that $x \mapsto x^n$ is increasing on $(0,+\infty)$. So in fact $x_n =1$ for all $n$.

Answer 2

for $n>0$, $x\mapsto x^n $ is increasing at $(0,1) $ thus

$$x_n=\sup_{x\in(0,1)}\{x^n\}=\lim_{x\to1^-}x^n=1$$

and $$\lim_{n\to+\infty}x_n=1$$