# Metric Spaces

#### tomboi03

Let X be a set, and let fn : X $$\rightarrow$$ R be a sequence of functions. Let p be the uniform metric on the space Rx. Show that the sequence (fn) converges uniformly to the function f : X $$\rightarrow$$ R if and only if the sequence (fn) converges to f as elements of the metric space
(RX, p)

I'm not sure how to do this problem...

Can someone help me out?

Thanks!

#### yyat

Where are you stuck? Start by writing down all the relevant definitions (uniform metric, uniform convergence, convergence in metric spaces) and you will be almost done.

#### de_brook

please define what is mean't by " uniform metric"

#### yyat

please define what is mean't by " uniform metric"
It's the metric obtained from the http://en.wikipedia.org/wiki/Uniform_norm" [Broken], so

$$d(f,g)=\sup_{x\in X}|f(x)-g(x)|$$

where $$f,g:X\to\mathbb{R}$$.

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