Uniformly continuous functions

The question:

Suppose $\displaystyle g(a,b] \rightarrow \mathbb{R}$ is uniformly continuous. Show that if {$\displaystyle x_n$} is a Cauchy sequence in (a,b], then {$\displaystyle g(x_n)$} is always a Cauchy sequence.

So I need to show that $\displaystyle \forall \epsilon >0 \exists N \in \mathbb{N} s.t. \forall m,n>N |g(x_m)-g(x_n)|<\epsilon$. I have then written out the definition for g being uniformly continuous on (a,b] but I can't see where to go from this definition to show the Cauchy sequence. Help?