8.2. Uniform Convergence
Example 8.2.2 (b): Pointwise vs Uniform Convergence
Let f_{n}(x) = x^{n} with domain
D = [0, 1]. Show that
{ f_{n}(x) } converges pointwise but not uniformly.
What if we change the domain slightly to D = (0, 1)?
Let Then:
Hence f_{n}(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 f_{n}(x) = x^{n} 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
f_{n}(x)  f(x) =  x^{n}  < 1/2 for all n > NThen in particular  x^{N+1}  < 1/2 for some fixed N. But if we now pick x such that
1 > x > (1/2)^{1/N+1}we have a contradiction.