limit of continuous functions

Question

If $f,g:[a,\infty)\to \Bbb R$ are continuous functions and $f$ is uniformly continuous on $[a,\infty)$ and $$\lim_{x\to \infty} (f(x)-g(x)) = 0,$$ how can I show that $g$ is uniformly continuous on $[a,\infty)$?
Because $f,g$ are continuous, can we assume that the limit of sums is the sum of the limits? Thanks.

Answer 1

Hint: You know that you can make $\lvert f(x)-g(x)\rvert$ small for all $x\geq N$ by choosing $N$ sufficiently large, because $f(x)-g(x)\to0$ as $x\to\infty$.

Now, $g$ is uniformly continuous on $[a,2N]$, because it is continuous and $[a,N]$ is compact. So, what remains is to show that you are also uniformly continuous on, say, $[N,\infty)$. (I say to use $[a,2N]$ and $[N,\infty)$ because using $[a,N]$ and $[N,\infty)$, you have to worry about the case where you have points close to the boundary.)

So, you need to show that $g$ is uniformly continuous on $[N,\infty)$. But, you know that $f$ is uniformly continuous on $[N,\infty)$... and that $f$ and $g$ do not differ by very much there.

Does that show you the heuristic reason for this to be true? That heuristic translates fairly directly in to a proof.