Convergence of summable sequences If $(a_n)$ is a sequence such that
$$\lim_{n\to\infty}\frac{a_1^4+a_2^4+\dots+a_n^4}{n}=0.$$
How do I show that $\lim_{n\to\infty}\dfrac{a_1+a_2+\dots+a_n}{n}=0$?

Since:
$$\left|\sum_{k=1}^{n} a_k\right|\leq \sqrt{n}\sqrt{\sum_{k=1}^{n} a_k^2}\leq n^{\frac{3}{4}}\left(\sum_{k=1}^{n}a_k^4\right)^{\frac{1}{4}}$$
by applying the Cauchy-Schwarz' inequality twice (or the Holder's inequality once), we have:
$$\frac{1}{n}\left|\sum_{k=1}^{n}a_k\right|\leq\left(\frac{1}{n}\sum_{k=1}^{n}a_k^4\right)^{\frac{1}{4}}, $$
so, if the RHS tends to zero, so does the LHS.
