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

show that f+g is uniformly continuous on X

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