Limit of Sequence of Supremums

Question

I just wanted to sanity check: $\text{sup}_{x\in[0,1)}\{|x^n|\}$ is a decreasing sequence, right? And so $\text{lim}_{n\to\infty}\text{sup}_{x\in[0,1)}\{|x^n|\}$ would just be some $x^n\neq 0?$ Thanks!

Answer 1

Not quite. For each $n\in\Bbb N$, $\sup_{x\in[0,1)}|x^n|=1$, and therefore $\lim_{n\to\infty}\sup_{x\in[0,1)}|x^n|=1$.