Suppose that is continuous and that is such that pointwise.
Show that uniformly if and only if for every sequence such that , we have that .
My work: I've finished the forward implication. This is my work on the reverse. We'll prove the contrapositive.
Suppose that does not converge to uniformly. So, there is some so that for any , there is an and an so that .
So, is a bounded sequence of reals, and so has a convergent subsequence, say such that
Since is continuous at , we can find an such that for any ,
But then, if , we have that
and so does not converge to .
Something about this argument feels incredibly wrong, and I was wondering if anyone could point out where I went awry.