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

or

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.