Give an example of a equicontinuous that does not converge uniformly

Question

Give an example of a equicontinuous sequence of functions ($f_n$) over a non-compact set $S\subset\Bbb R^n$ converging pointwise to a function $f$ at each $x\in S$, but $f_n$ does not converge uniformly to $f$ over $S$.

I'm really stuck on this problem, and I thought about the cases of $f_n(x) = x^n$ with the domain ($0,1$) or $f_n(x) = sin(nx)$ over non-compact set, but I failed to derive an example. Could someone help me to find an example please? Thanks

Answer 1

Take the functions $f_n(x)=x/n$ over $\Bbb R$, converging non-uniformly to $f(x)=0$.

Answer 2

$$f_1(x) = 0$$ except in a "hump" between 0 and 1. $$f_n(x) = f_1(x-n).$$