## Example 8.2.6 (a): Uniform Convergence and Sup Norm

Consider the sequence

*f*:_{n}(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
f_{n}'does notconverge 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:

The sequence of derivatives is||f_{n}- f||_{D}= ||1/n sin(n x) - 0||_{D}1/n 0

which does not converge (take for examplef '(x) = cos(n x)

*x =*).