Prove that if $f: [a, \infty) \rightarrow \Bbb R$ is continuous on $[a, \infty)$ and has a finite limite $ \lim_{x\to \infty} f(x)$, then f is uniformly continuous.
I found such definition of uniform continuity: $\forall x_n,y_n \subset [a, \infty) \lim_{x\to \infty} x_n-y_n=0 \Rightarrow \lim_{x\to \infty} f(x_n)-f(y_n)=0$
and tried to do this using Heine's definitions but i got stuck at one point and can't move on. I'd really appreciate any hints!