Show uniform continuity

Question

I hope anyone can help with this!

Assume that

$f(x) \rightarrow a$ for $x \rightarrow \infty$

$f(x) \rightarrow b$ for $x \rightarrow -\infty$.

I've already shown that for every $\epsilon>0$ there exists an $M>0$ such that $|f(x)−f(y)|<\epsilon$ as long as $x,y>M$ and now I have to show that $f$ is uniformly continuous but I don't know how and would appreciate some inputs.

Thanks a lot!

Answer 1

Hint:f is continous on [-2M,2M], which is a compact se so is uniformly continous.