Interactive Real Analysis

• Real Analysis
• 1. Sets and Relations
• 2. Infinity and Induction
• 3. Sequences of Numbers
• 4. Series of Numbers
• 5. Topology
• 6. Limits, Continuity, and Differentiation
• 7. The Integral
• 8. Sequences of Functions
• 8.1. Pointwise Convergence
• 8.2. Uniform Convergence
• 8.3. Series and Power Series
• 8.4. Taylor Series
• 8.5. Approximation Theory
• 9. Historical Tidbits
• Java Tools

8.2. 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.