Let f, g : D→R be uniformly continuous. Prove that f-g: D→R is uniformly continuous aswell
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.
For all ε >0 there exists δ >0 s.t.
abs( x - y ) < δ => abs( f(x) - f(y)) <ε
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.