Example 8.1.8 (d): Pointwise Convergence does not inherit Derivatives
fn(x) = 1/n sin(n x)
Then | fn(x) | 1/n for all x so that the sequence converges to zero for all x. Each fn(x) is differentiable with
f 'n(x) = cos(n x)
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.