Given the theorem: Suppose $S \subset \Bbb R^n$ is compact, and $P$ is an equicontinuous sequence of functions ($f_n$) over $S$ converging pointwise to a function $f$ at each $x \in S$, then $f_n$ converges to $f$ uniformly over $S$.
Note that the definition here is: $Q$ as a set of real-valued functions defined over $S \subset \Bbb R^n$ is equicontinuous at $y \in S$ if given $\epsilon \gt 0$ there is a $\delta \gt 0$ such that $|f(x)-f(y)|\lt \epsilon$ for all $|x-y|\lt \delta$ and all $f \in Q$. Also, it is equicontinuous over $S$ if it is equicontinuous at all $x \in S$
I want to give an example to show that the above theorem fails if $S$ is not compact but stuck on finding one. Could someone provide an example please？Thanks a lot.