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

Question

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.

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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.

Answer 1

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.

Answer 2

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.

Answer 3

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.