Does the sequence of functions $[f(t)]^n$ converge uniformly to $0$ if $|f|<1$ on $[a,b]$?

Question

Suppose that $f$ is continuous and $|f(x)|<1$ for all $x$ in $[a,b]$. Does $[f(t)]^n$ converge uniformly to $0$?

Is it a no? Because if I have $x^n$ then at $x$ very close to $1$ it takes very long to converge to $0$, so we can't find an $N$ such that with all $n > N$, $x^n$ converges to $0$.

Answer 1

HINT: The function $|f|$ is continuous, so it attains its maximum at some point $c\in[a,b]$. Clearly $0\le |f(c)|<1$. What can you say about $|f(x)|^n$ in terms of $|f(c)|$?

Answer 2

It does, since $f$ is continuous $|f|$ is continuous. Now use the fact a continuous function on a bounded closed interval reaches a maximum. so let $M$ be the maximum of $|f|$ on $[a,b]$.

We then have $|f^n(x)|<M^n$ for all $x\in[a,b]$. Since $M<1$ we have $\lim\limits_{n\to \infty}M^n=0$. And so $f^n(x)$ converges uniformly to the function $0$.