Question

In: Advanced Math

For each n ∈ N, let fn : [0, 1] → [0, 1] be defined by...

For each n ∈ N, let f_n : [0, 1] → [0, 1] be defined by f_n(x) = 0, x > 1/n and f_n(x) = 1−nx if 0 ≤ x ≤1/n.

The collection {f_n(x) : n ∈ N} converges to a point, call it f(x) for each x ∈ [0, 1]. Show whether {f_n(x) : n ∈ N}
converges to f uniformly or not.

Expert Solution

for all n. For by Archimedean property real numbers, there exist (a fixed natural number) such that . So for all

. Hence, for all , and (by definition). This is because for, after finitely many non-zero all other . So, and as at , it is the constant sequence 1and for , after finitely many non-zero terms, the sequence is eventually constant sequence 0.

Claim: does not converges to uniformly.

where supremum runs over . Since, 1 does not tends to zero. Hence, we are done.

Also, there is a theorem that if sequence of continuous functions converges uniformly then the limit function is continuous. Here, are continuous throughout and f(x) is not continuous at x=0. So, they cannot converge uniformly.

Colby Messinger answered 6 months ago

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