Equivalent conditions for bounded sequence of non-negative real numbers

Let (an)(a_n) b a bounded sequence of non-negative real numbers. Then the following are equivalent:

1) limn→∞1n∑n−1j=0aj=0\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1} a_j=0,

2) \lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1} a_j^2=0\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1} a_j^2=0.

How to prove this? I don’t even know from where to start.

So, any help or hint is appreciated. Thanks in advance.




If a_n \in [0,b]a_n \in [0,b], then a_n^2 \leqslant b\cdot a_na_n^2 \leqslant b\cdot a_n. For the other direction, use Jensen’s inequality or Cauchy-Schwarz.
– Daniel Fischer♦
2 days ago