8.2. Uniform Convergence

Example 8.2.6 (a): Uniform Convergence and Sup Norm

Consider the sequence fn(x) = 1/n sin(n x):
  • Show that the sequence converges uniformly to a differentiable limit function for all x.
  • Show that the sequence of derivatives fn' does not converge to the derivative of the limit function.

This example is ready-made for our sup-norm because |sin(x)| < 1 for all x. As for our proof: the sequence converges uniformly to zero because:

||fn - f||D = ||1/n sin(n x) - 0||D 1/n 0
The sequence of derivatives is
f '(x) = cos(n x)
which does not converge (take for example x = ).

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