MathCS.org - Real Analysis: Example 8.2.2 (d): Pointwise vs Uniform Convergence

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?

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Not true. We already met the sequence f_n(x) = max(n - n² |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) = xⁿ on [0, 1] would be another case in point.

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Example 8.2.2 (d): Pointwise vs Uniform Convergence