8.2. Uniform Convergence

Example 8.2.2 (a): Pointwise vs Uniform Convergence

Define fn(x) = x/n with domain D = [a, b]. Show that the sequence { fn(x) } converges uniformly to zero. What if we change the domain to all real numbers?

Take any > 0. Let N = max(|a|, |b|)/. Then

|fn(x) - 0| = |x/n| max(|a|, |b|) / n <

as long as n > N. Note that because we use the max function, N does not depend on x. It does depend on a and b (and of course on ) but those are fixed numbers.

If we change the domain to all real numbers, the sequence still converges pointwise to zero, but it no longer converges uniformly, because if it did, then for, say, = 1 there would have to be an N such that:

|x/n| < 1 for all n > N

But then

|x| < n for all n > N

which implies

|x| < N+1

Since x is unbounded, we have a contradiction.

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