Been reading through this proof which seems incorrect:
Let $f_n$ be continuous on the curve $C$ and $\sum f_n$ converge uniformly on $C$. Then $\sum\int_Cf_n(z)dz=\int_C\sum f_n(z)dz$
PROOF: $f:=\sum f_n$ and $L:=$ length of $C$. Then $$\left|\int_C f(z)dz-\int_C\sum^m_{n=0}f_n(z)dz\right|\leq L\: \max_{z\in C}\left| f(z)-\sum^m_{n=0}f_n(z) \right|\rightarrow0$$
It seems to me that this does not prove anything at all as we're still summing inside the integral. Where am I/this proof wrong?
Thank you.
In the proof you are just swapping a finite sum and integral, the theorem is about moving up the result from finite sums to infinite uniformly convergent sums.