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## Homework Statement

Let f, g : D→R be uniformly continuous. Prove that f-g: D→R is uniformly continuous aswell

## Homework Equations

none

## The Attempt at a Solution

Okay, I am posting this question because I want to make sure that my solution is correct and if it isn't I would really be thankful if someone pointed out its flaws.

My solution:

For all ε >0 there exists δ >0 s.t.

abs( x - y ) < δ => abs( f(x) - f(y)) <ε

then

abs( f(x) - f(y)) <ε/2

abs( g(x) - g(y)) <ε/2

now its f - g so

abs( f(x) - f(y) - g(x) + g(y)) < ε

abs( f(x) - f(y)) + abs( - g(x) + g(y)) < ε

abs( f(x) - f(y)) + abs(-1)*abs( +g(x) -g(y)) < ε

abs( f(x) - f(y)) + abs( +g(x) -g(y)) < ε

ε/2 + ε/2 <ε

ε ≤ ε

Also can anyone describe to me what uniformly continuous means besides from the actual definition. I have a hard time understanding what uniformly continuous means.

Thanks.