Suppose $f_n$ are continuous functions converging uniformly to $f$ on the compact interval $[a,b]$.
a) Show that for any natural power $k$, the sequence $f_n^k$ converges uniformly to $f^k$ on [a,b].
b) If $p$ is a polynomial, show that $p \circ f_n$ converges uniformly to $p \circ f$ on $[a,b]$.
For part a) We know that, since the sequence is uniformly convergent,
$$\mathrm{sup}||f_n – f ||=M_n \rightarrow 0$$
Now can I say,
$$\mathrm{sup}||f_n^k – f^k || = M_n^k\rightarrow 0$$
and hence it is uniformly convergent. I am a bit sceptical about this.
For b) I think all polynomials are continuous and bounded on an interval and hence uniformly convergent. Hence $p \circ f_n$ is uniformly convergent to $p \circ f$. But I don't know how to exactly prove it.
Any help would be greatly appreciated.
Best Answer
For a) see also The sequence $f_n^k$ converges uniformly to $f^k$ on $[a,b]$
For b), note that a polynomial is a linear combination of powers. Use the triangle inequality and a) ,
Bonus question: prove the following more general result. Let $g:A\to \mathbb{R}$ be a Lipschitz function and $f_n,f:[a,b]\to A$. If $f_n$ converges uniformly to $f$ in $[a,b]$ then $g\circ f_n$ converges uniformly to $g\circ f$ in $[a,b]$.