Let X be a set, and let f(adsbygoogle = window.adsbygoogle || []).push({}); _{n}: X---> R be a sequence of functions. Let ρ be the uniform metric on the space R^{X}. Show that the sequence (f_{n}) converges uniformly to the function f:X--> R if and only if the sequence (f_{n}) converges to f as elements of the metric space (R^{X}, ρ). [Note: the ρ's should have a bar over them.]

I have a question concerning one direction of our implication.

[My attempt at the solution.]

Since I do not know how to make a d with a bar over it, I'll use ∂ to denote the standard bounded metric relative to d.

Proof:

Assume the sequence (f_{n}) converges to f as elements of the metric space (R^{X}, ρ).

Then, for any ε>0, there is a positive integer N such that ρ(f_{n},f)<ε for all n≥N.

That is, for n ≥ N, the sup{∂(f_{n}(x),f(x))| x in X} < ε.

If ε≤1, our metrics are the same.

Hence for ε≤1, there exisits an integer N such that for n ≥ N, the sup{∂(f_{n}(x),f(x))| x in X} < ε.

But since ∂(f_{n}(x),f(x)) = d(f_{n}(x),f(x)) for n ≥ N, d(f_{n}(x),f(x)) ≤ sup{∂(f_{n}(x),f(x))| x in X} < ε for all n≥N and x in X, which satisfies our condition for uniform convergence.

My question is about ε. Is this the right approach?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Uniform Convergence and the Uniform Metric

**Physics Forums | Science Articles, Homework Help, Discussion**