Let $0<b<1$ and $f_n(x)=x_n$ defined on $[0,b]$. Show that $f_n\to0$ uniformly on $[0,b]$

Question

Let $0<b<1$ and $f_n(x)=x_n$ defined on $[0,b]$. Show that $f_n\to0$ uniformly on $[0,b]$.

I tried almost everything to prove the statement. I tried to construct a sequence $x_{n_k}$ which is convergent (so Cauchy) as $x_n$ is bounded on $[0,b]$ and to apply uniform continuity of $f_n$ but I can't conclude the proof. If someone could give a hint, I would really appreciate it. Thank you in advance.

Answer 1

If $f_n(x)=x^n$, then $$\lim_{n\to \infty} \sup_{x\in [0,b]} |f_n(x)-0|=\lim_{n\to \infty}b^n=0,$$ since $0<b<1$.

EDIT: If $b\ge 1$, then $\lim_{n\to \infty}b^n\neq 0$ and the convergence is not uniform.