Study the Limit of a Sequence of Functions

This example illustrates how the failure of uniform convergence for a sequence of functions leads to non-commutativity of limiting operations applied to the sequence.

The following sequence of functions converges to zero at every point .

However, the maximum value of at the point diverges as .

This shows that the convergence of the sequence of functions is not uniform.

As a consequence, the limit of the integrals does not equal the integral of the limit.