Real Analysis/Uniform Convergence
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Definition: a sequence of real-valued functions fn(x) is uniformly convergent if there is a function f(x) such that for every ε>0 there is an N>0 such that when n>N for every x in the domain of the functions f, that |fn(x)-f(x)|<ε
[edit] Theorem (Uniform Convergence Theorem))
Let fn be a sequence of continuous functions that uniformly converges to a function f. Then f is continuous.
[edit] Proof
There exists an N such that for all n>N, 
for any x. Now let n>N, and consider the continuous function fn. Since it is continuous, there exists an δ such that if | x' − x | < δ, then 
. Then 
so the function f(x) is continuous.
