Let $(a_n)_{n\in \mathbb N}$ be a bounded non-negative sequence. Show that $$\lim_{n\rightarrow \infty}\dfrac{1}{n}\sum_{k=0}^{n-1} a_k = 0\tag{1}$$ if and only if $$\lim_{n\rightarrow \infty}\dfrac{1}{n}\sum_{k=0}^{n-1}a_k^2= 0. \tag{2}$$
I am aware of the following characterization of (1), and this question on MSE:
(1) if and only if there exist a subset $J\subset \mathbb N$ with density zero, such that $(a_n)\xrightarrow[n\notin J]{n\rightarrow \infty}0$.
How can I use this to prove (2)?
Im thinking for (1) implies (2), one can say that for the same subset $J\subset \mathbb N$, one has $$(a_n^2)\xrightarrow[n\notin J]{n\rightarrow \infty}0.$$
As for (2) implies (1), i can repeat the proof for constructing the set of density zero for the series $\dfrac{1}{n}\sum_{k=0}^{n-1}a_k^2$ instead, and use it to characterize (1).
Is my reasoning correct or am i missing something that is obvious?
$a_n \to 0$ through $J^{c}$ (which means $n$ is restricted to integers not in $J$) iff $a_n^{2} \to 0$ through $J^{c}$ and this gives the equivalence. [You have apply the characterization both the sequences $(a_n)$ and $(a_n^{2})$. You can take the same $J$ for both of these because union of two sets of density $0$ has density $0$].