For any positive integers define Evaluate and
The function is in , and by the Riemann-Lebesgue lemma . Therefore .
The function is in (with ) and is improperly Riemann integrable there, so .
The function (with ) is in , and is improperly Riemann integrable and Lesbesgue integrable there.
By the Riemann-Lebesgue lemma, .
Thus . Therefore .