Example of limit supremum of functions not equal to supremum of limit

Question

I am trying to figure out a counterexample where limit supremum of functions is not equal to supremum of limit.

Let $f_n: E \to \mathbb{R}$ , $\lim_{n \to \infty} \sup_{x \in E}f_n(x) =\sup_{x \in E} \lim_{n \to \infty} f_n(x)$

Can anyone give a counter example?

Answer 1

Define $f_n\colon\Bbb R\to\Bbb R$ by $$f_n(x)\colon=\begin{cases}1&\text{ if }x\in \left(-\frac{1}{n},\frac{1}{n}\right),\\0&\text{ elsewhere. }\end{cases}$$ Then for each $n$ we have, $$\sup_{x\in \Bbb R}f_n(x)=1.$$ Also, for each $x\in \Bbb R$ we he have, $$\lim_{n\to \infty}f_n(x)=0.$$ So, $$\lim_{n \to \infty} \sup_{x \in E}f_n(x)=1 \not=0=\sup_{x \in E} \lim_{n \to \infty} f_n(x)$$