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

Remember that we discussed
uniform continuity in a previous chapter.
We showed that a function that is (regularly) continuous on a compact set is
automatically uniformly continuous.
Is that true also for pointwise and uniform convergence, i.e. is a sequence that
converges pointwise on a compact set automatically uniformly convergent?

Not true. We already met the sequence
*f _{n}(x) = max(n - n^{2} |x - 1/n|, 0)*
which converges pointwise to zero on the closed, bounded (i.e. compact)
interval

*[0, 1]*but not uniformly.

The sequence *f _{n}(x) = x^{n}* on

*[0, 1]*would be another case in point.