Your logic is correct but your writing is misleading. When you say "bythe definition of uniform continuity" you are signalling that you're about to use uniform continuity as a hypothesis. But instead you use Lipschitz continuity---which is what you should do!

A proper write-up would look something like this...

Let \to R" alt="f\to R" /> satisfy a Lipschitz condition with constant , and let . Then there is such that for any , if then . Hence is uniformly continuous.