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.

If $f_n$ are not continuous then it is not true.
Take  $f_n=1_{[-\frac{1}{2},\frac{1}{2}]}+\frac{1}{n}$ on $[-1,1]$
where $1_A$ is the indicator function of the set  $A$
You can see in many textbooks and notes that: If the  functions $f_n$ are continuous on a compact interval ,then their uniform limit is continuous on that interval.
