## Theorem 8.2.7: Uniform Convergence and Integration

Let

*f*be a sequence of continuous functions defined on the interval_{n}(x)*[a, b]*and assume that*f*converges uniformly to a function_{n}*f*. Then*f*is Riemann-integrable andf_{n}(x) dx = f_{n}(x) dx = f(x) dx

Since *f _{n}* are continuous and converge uniformly to

*f*, the limit function must be continuous. In particular all functions must therefore be Riemann integrable. Also:

Since the right side goes to zero as

*n*goes to infinity we are done.