Uniform convergence of a Cauchy Sequence in a compact implies continuous limit. 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.

Without continuity of $f_n$'s this is false. Take any bounded function $f$ which is not continuous and let $f_n =f $. Then $f_n \to f$ uniformly. 
