If $f:\Bbb{R}\to\Bbb{R}$ is uniformly continuous, then $f$ is bounded

Question

I have proved :

If $f$ is uniformly continuous on $(a,b)$, then $f$ is bounded on $(a,b)$.

But when it extends $(a,b)$ to $\Bbb{R}$, I don't know how to continue. Would you please give some instructions?

Answer 1

That is not true for the whole real line but it is true for any closed interval. It is in fact true for any compact subset of real numbers.