Unfirom convergence of sequence of functions Let $\{f_n(x)\}$ be sequence of functions defined on a set $E$. Let for any $n$ we have $|f_n(x)|\leqslant c_n$ and $c_n\to 0$ as $n\to \infty$. Does it follow that $f_n$ converges uniformly on $E$?

I can't prove it rigorously.
Yes,

By the given hypothesis if $f(x)=\lim_{n\to \infty} f_n(x)$ then $f(x)=0$.

So $M_n=\sup_{x\in E} |f_n(x)-f(x)|=\sup_{x\in E} |f_n(x)|\leq c_n\to 0$ for $n\to \infty $

Thus $f_n$ converges uniformly to $f$
Counterexample is the following:

Let $f_n(x)=x^n$ on $E=[0,1)$. For any fixed $n$ we see that $|f_n(x)|=x^n&lt;1=c_n$ and $c_n\to 1\neq 0$ but $\{f_n(x)\}$ does not coverges uniformly on $E$.