Let $f:[0,\infty) \rightarrow \mathbb R$ be a continuous function. Define $f_n:[0,\infty) \rightarrow \mathbb R$ by
$$f_n(x) := f(x+n).$$
If $f_n$ converges uniformly to $g$, then f and g is uniformly continuous.
My first atttempt is as follows. By triangle ineqality, it holds
$$ |g(x) - g(y)| \leq |g(x) - f_n(x)| + |f_n(x) - f_n(y)| + |f_n(y) - g(y)|.$$
The first and third can be made as small as possible. But, how about the second term?
$g(x+1)=\lim f(x+n+1)=\lim f(x+n)=g(x)$ so $g$ is continuous (by uniform convergence) and periodic. Hence $g$ is unifromly continuous.
Hint for uniform continuity of $f$: $|f(x+n)-f(x+m)| <\epsilon$ for all $x$ if $n,m$ are suffciently large. This shows that for some $T>0$ we have $|f(x)-f(x+k)| < \epsilon$ for $x>T$ for all $k$. Can you finish?