Real Analysis/Uniform Convergence

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Real Analysis
Uniform Convergence

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

Theorem (Uniform Convergence Theorem))[edit]

Let f_n be a series of continuous functions that uniformly converges to a function f. Then f is continuous.

Proof[edit]

There exists an N such that for all n>N, |f_n(x) - f(x)|<\frac{\epsilon}{3} for any x. Now let n>N, and consider the continuous function f_n. Since it is continuous, there exists a \delta such that if |x'-x|<\delta, then |f_n(x)-f_n(x')|<\frac{\epsilon}{3}. Then |f(x')-f(x)|\le |f(x')-f_n(x')|+|f_n(x')-f_n(x)|+ |f_n(x)-f(x)| < \frac{\epsilon}{3}+\frac{\epsilon}{3}+\frac{\epsilon}{3} = \epsilon so the function f(x) is continuous.