## Example 8.2.2 (a): Pointwise vs Uniform Convergence

Define

*f*with domain_{n}(x) =^{x}/_{n}*. Show that the sequence**= [a, b]***D***{ f*converges uniformly to zero. What if we change the domain to_{n}(x) }*all real*numbers?Take any * > 0*. Let
*N = max(|a|, |b|)/*. Then

|f_{n}(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| < 1for alln > N

But then

|x| < nfor alln > N

which implies

|x| < N+1

Since *x* is unbounded, we have a contradiction.