# On the convergence of a sequence of functions

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 $$E$$?

• Use Heine-Cantor to conclude that $g$ is uniformly continuous on $[-M,M]$. – logarithm May 8 at 12:05
• no idea actually! need some more help – Tharindu Rasanga May 8 at 12:11
• 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 $x\in E$. – logarithm May 8 at 12:21
• thanks for the help – Tharindu Rasanga May 8 at 13:04