Example 8.2.2 (b): Pointwise vs Uniform Convergence
Hence fn(x) f(x) pointwise for each fixed x.
Uniform convergence on [0, 1] will fail, just by looking at the picture, because the difference between f(1)=1 and fn(x) = xn for x < 1 will get larger and larger. In fact, it won't matter if we take the closed interval [0, 1] or the open one (0, 1) because:
Take, say, =1/2 and let x < 1. Assume there exists an integer N such that
|fn(x) - f(x)| = | xn | < 1/2 for all n > NThen in particular | xN+1 | < 1/2 for some fixed N. But if we now pick x such that
1 > x > (1/2)1/N+1we have a contradiction.