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.