9.3 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 9.3.1 Let g be of bounded variation and let fn be continuous and
converging uniformly to f on
. Then it follows f is also integrable and
Proof: The uniform convergence implies f is also continuous. See Theorem 6.9.7.
abfdg exists. Now let n be given large enough that
Next pick δ > 0 small enough that if
for any choice tk ∈
. Pick such a
and the same tk
for both sums. Then
Since ε is arbitrary, this shows that limn→∞