Let be an integrable and uniformly continuous function. Prove that.
Solution
This exercise is quite known. It first made its appearance as a Berkley exam question.
Suppose that the limit is not zero. Then let us pick a sequence and an such that foreach to both hold and . Since is uniformly continuous there exists such that
However in the interval it holds that
foreach positive. The last is a consequence of . And of course this contradicts the fact that the function is integrable. Hence the conclusion.
Note: We can’t drop the condition of uniform continuity. If so, then the function is a counterexample. Indeed it is continuous, integrable but the limit at infinity is not .