Interactive Real Analysis
- part of Example 8.2.2 (a): Pointwise vs Uniform Convergence
Define fn(x) = x/n with domain
D = [a, b]. Show that the sequence
{ fn(x) } converges uniformly to zero. What if
we change the domain to all real numbers?
Back
BackTake any 
> 0. Let
N = max(|a|, |b|)/. Then
|fn(x) - 0| = |x/n|max(|a|, |b|) / n <
as long as n > N. Note that because we use the max
function, N does not depend on x. It does
depend on a and b (and of course on
) but those are fixed numbers.
If we change the domain to all real numbers, the sequence still converges
pointwise to zero, but it no longer converges uniformly, because if it did,
then for, say, = 1 there would
have to be an N such that:
|x/n| < 1 for all n > N
But then
|x| < n for all n > N
which implies
|x| < N+1
Since x is unbounded, we have a contradiction.