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Suppose that $f_n → f$uniformly on some set $E$ and that for each $n$, there exists $M_n$ such that

$$|f_n(x)| ≤ M\quad\text{for all }n=1,2,3...\text{ and all }x ∈ E.$$

Suppose $g$ is a continuous function on $[-M,M]$.

How to show that $g(f_n(x))$ converges uniformly to $g(f(x))$ on $E$?

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  • $\begingroup$ Use Heine-Cantor to conclude that $g$ is uniformly continuous on $[-M,M]$. $\endgroup$ – logarithm May 8 at 12:05
  • $\begingroup$ no idea actually! need some more help $\endgroup$ – Tharindu Rasanga May 8 at 12:11
  • $\begingroup$ Let $\epsilon>0$. Since $g$ is uniformly continuous on $[-M,M]$ then there is $\delta>0$ such that for every $r,s\in[-M,M]$ with $|r-s|<\delta$ it will happen that $|g(r)-g(s)|<\epsilon$. Now, since $f_n$ converges uniformly, there is $N$ such that for every $n>N$ and all $x\in E$ you will have $|f_n(x)-f(x)|<\delta$. Since $f_n(x)$ and $f(x)$ are both in $[-M,M]$ we can take $r=f_n(x)$ and $s=f(x)$ above, giving $|g(f_n(x))-g(f(x))|<\epsilon$ for all $n>N$ and $x\in E$. $\endgroup$ – logarithm May 8 at 12:21
  • $\begingroup$ thanks for the help $\endgroup$ – Tharindu Rasanga May 8 at 13:04

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