1. ## Cauchy sequence/Continuity Question

Let $\displaystyle f: (a,b] \rightarrow \mathbb{R}$ be a real valued continuous function defined on the half open interval (a,b]. Assume that f has the property than whenever $\displaystyle (x_n)$ is a cauchy sequence contained in the open interval (a,b], then $\displaystyle (f(x_n))$ is a cauchy sequence.
Prove that the one sided limit:
$\displaystyle \lim_{x \to a+} f(x)$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?

2. ## Re: Cauchy sequence/Continuity Question

Originally Posted by worc3247
Let $\displaystyle f: (a,b] \rightarrow \mathbb{R}$ be a real valued continuous function defined on the half open interval (a,b]. Assume that f has the property than whenever $\displaystyle (x_n)$ is a cauchy sequence contained in the open interval (a,b], then $\displaystyle (f(x_n))$ is a cauchy sequence.
Prove that the one sided limit:
$\displaystyle \lim_{x \to a+} f(x)$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?
The idea is to prove that you can continuously extend $\displaystyle f: (a,b]$ to $\displaystyle \widetilde{f}:[a,b]$ by defining $\displaystyle \displaystyle f(a)=\lim_{x\to a^+}f(x)$. You can then conclude from the Heine-Cantor theorem that $\displaystyle \widetilde{f}$ is unif. cont. and since the restriction of unif. cont. maps are unif. cont. you'd have that $\displaystyle f$ is unif. cont.