Uniformly continuous $f$ in $L^p([0,\infty))$ Assume that $1\leq p < \infty $,  $f \in L^p([0,\infty))$, and $f$ is uniformly continuous. Prove that $\lim_{x \to \infty} f(x) = 0$ .

Hints: Suppose for a contradiction that $f(x) \not \to 0$. Together with the definition of uniform continuity, conclude that there exist constants $\varepsilon > 0$ and $\delta = \delta(\varepsilon) > 0$ such that for any $M > 0$ there exists $x > M$ for which it holds
$$
\int_{(x,x + \delta )} {|f(y)|^p \,dy}  \ge \bigg(\frac{\varepsilon }{2}\bigg)^p \delta .
$$
However, this would imply $\int_{[0,\infty )} {|f(x)|^p \,dx}  = \infty$, a contradiction. 
