This has the definition you need. Right at the top.
"Prove that if a series of continuous functions converges uniformly, then the sum function is also continuous."
I'm trying to interpret what this is saying so that I can know what to prove. Does the given portion mean that the individual f_n's uniformly converge to f, or that ?
And what does "the sum function is continuous" mean? Simply that is continuous?
This has the definition you need. Right at the top.
OK. Can we just use the fact that if (1) f_n is continuous for all n's, and (2) f_n -> f uniformly, then (3) f is continuous? I.e., does the result I'm trying to prove immediately follow, since the sum of continuous functions on an interval is also continuous on that interval?
Yes you can.
Use what Ken Ross calls the famous $\dfrac{\varepsilon }{3}>0$ argument.
$|f(x)-f(x_0)|\le |f(x)-f_n(x)|+|f_n(x)-f_n(x_0)|+|f_n(x_0)-f(x)| $
You select an $n$ large enough to insure that on RHS the first and third term is less that the magic number from the continuity of $f_n$.
The middle comes from the uniform convergence of the sequence.
Thanks, Plato, but I think you have a few errors in there. Isn't that last term supposed to be f_n(x_0) - f(x_0)? And uniform convergence produces the first and third terms, not the second one, right?