6.3 Uniform Convergence And The Integral
It turns out that uniform convergence is very agreeable in terms of the integral. The
following is the main result.
Theorem 6.3.1 Let fn be continuous and converging uniformly to f on
Then it follows f is also continuous and
Proof: The uniform convergence implies f is also continuous. See Theorem 4.9.3.
abfdx exists. Using the triangle inequality and definition of
earlier in conjunction with this theorem,
which is given to converge to 0 as n →∞. ■