If then from uniform convergence.
Now from uniform continuity .
24.13 Prove that if (fn) is a sequence of uniformly continuous functions on an interval (a,b), and if fn -> f uniformly on (a,b), the f is also uniformly continuous on (a,b). Hint use eps/3 argument.
I don't see what I need to change from the "limit of continuous functions is continuous" theorem. At the end of that proof it concludes f is continous at x0. I guess I'm not sure how to expand that to uniform continuity.