8.2. Uniform Convergence

Example 8.2.2 (b): Pointwise vs Uniform Convergence

Let fn(x) = xn with domain D = [0, 1]. Show that { fn(x) } converges pointwise but not uniformly. What if we change the domain slightly to D = (0, 1)?

Let

Then:

  • if x = 1 we have fn(1) = 1 for all n
  • if x < 1 then fn(x) = xn is the power sequence and thus converges to zero

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 > N
Then in particular | xN+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.
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