I need to show that if I have a Cauchy sequence of functions $f_n$.
which I know converges uniformly to $f$ in a compact set $[-r,r]$, then our limit function $f$ is continuous.
The thing is, for my definition of Cauchy sequence, it follows that a sequence is Uniformly Convergent iff it is a Cauchy sequence.
So maybe the question would not change if I wanted to show every uniformly convergent sequence in a compact converges to a continuous function.
But anyways I don't see how to procede because, for me, it would make a lot more sense if I knew the function $f_n$ to be continuous. In this case I could argue that they are uniformly continuous and I' know at least how to start.
Best Answer
You are correct to have doubts. For a counterexample to the statement just take each $f_n=f$, where $f$ is any discontinuous function of your choice. Then clearly $f_n\to g$ uniformly, and $(f_n)$ is uniformly Cauchy.
Note that to talk about cauchyness of a sequence of functions is generally ambiguous, unless it is known what function space you are working in. You can have a sequence of functions which are pointwise cauchy but not uniformly cauchy.