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.