Convergence Functions

Convergence of functions is introduced in Sequence of Functions | Real Analysis.

Uniform Convergence Theorem

Let be a sequence of Riemann integrable functions on . Suppose converges to uniformly. Then is Riemann integrable on and :

and:

Proof Hint

• Consider in place of .
• Consider Cauchy criteria for .
• Prove is Riemann integrable using Cauchy criteria.
• is Riemann integrable as .

When converges to pointwise, we cannot be sure if is Riemann integrable or not. An example where is not Riemann integrable:

Here is the enumeration of rational numbers in .

Dominated Convergence Theorem

Let be a sequence of Riemann integrable functions on . Suppose converges to pointwise where is Riemann integrable on . If :

Monotone Convergence Theorem

Let be a sequence of Riemann integrable functions on , and they are monotone (all increasing or decreasing, like ). Suppose converges to pointwise where is Riemann integrable on . If :

Can be proven from the dominated convergence theorem.