Proving Uniform Continuity for f+g on X

[CarmineCortez]

Homework Statement

if f and g are 2 uniformly continuous functions on X --> R

show that f+g is uniformly continuous on X

The Attempt at a Solution

I tried showing that f+g is Lipschitz because all Lipschitz functions are uniformly continuous.

So i end up with d(x_1,x_2) < delta then d([f(x1)+g(x1)],[f(x2)+g(x2)])<epsilon. how can I show this is lipshcitz

 

[jacobrhcp]

forget about lipschitz

you know that if you pick x1 and x2 close to each other, then fx1 and fx2 are at most a distance close to each other, namely epsilon. And also gx1 and gx2 are at most a distance epsilon. so their summed distance is 2 *epsilon.

so now you can define epsilon' as 2epsilon... so for any distance delta between x1 and x2 you now have a distance epsilon' between f+gx1 and f+gx2, that upholds that "for any epsilon' >0 there is a delta >0 such that the distance between x1 and x2 <delta implies that the distance between f+gx1 and f+gx2 is smaller than epsilon' "... it's just the same delta as before.

or at least that's what I make of it. It's 1 AM here and I can't finish my own problem so I went to look at other peoples' problems... but I don't have a clear mind right now.

Oh yeah, and when you write this down... you'll want to use nice math symbols and make things a bit more rigorous... but I think that should be no problem.

 

[Johnson04]

Let e>0 be given. Since f(x) is uniformly continuous on X, there exists s > 0, such that y in neighborhood of x, N_s (x), implies |f(y)-f(x)| < e/2. Similarly, since g(x) uniformly continuous, y in N_s (x) implies |g(y) - g(x)| < e/2. Since |f(y)+g(y) - f(x)-g(x)| <= |f(y) - f(x)| + |g(y) - g(x)| < e/2 + e/2, we have |(f+g)(y) - (f+g)(x)| < e, for all y in N_s (x). That is (f+g) uniformly continuous, this completes the proof.

 

[jacobrhcp]

I don't think we're supposed to write out the entire answer like that

 

[Dick]

I don't think we're supposed to write out the entire answer like that

VERY DEFINITELY NOT. It's against the forum rules. "Under no circumstances should complete solutions be provided to a questioner, whether or not an attempt has been made." Don't do that again.