Example 8.1.8 (d): Pointwise Convergence does not inherit Derivatives
Find a pointwise convergent sequence of differentiable functions whose
limit is differentiable but the sequence of derivatives does not converge.
Let f_{n}(x) = 1/n sin(n x) Then  f_{n}(x)  1/n for all x so that the sequence converges to zero for all x. Each f_{n}(x) is differentiable with f '_{n}(x) = cos(n x) Thus


Oops, I guess we should justify our claim that f '_{n}(x) = cos(n x) does not converge pointwise. Can you? As a hint, it would be enough to find one fixed x for which the resulting numeric sequence diverges.