Let be a real valued continuous function defined on the half open interval (a,b]. Assume that f has the property than whenever is a cauchy sequence contained in the open interval (a,b], then is a cauchy sequence.

Prove that the one sided limit:

exists and deduce that f is uniformly continuous.

So far all I have is the definition of a cauchy sequence and the definition of a right-hand sided limit, but can't see how to start this question. Help?