Finding the limit of a sequence.

Question

$(x_n)$ is a positive sequence and
$$\lim_{n\to\infty}\frac{x_n}{n}=0\hspace{6mm}\limsup_{n\to\infty} \frac{x_1+x_2+\ldots+x_n}{n}\in\mathbb{R}$$
and we need to find $$ \lim_{n\to\infty} \frac{x^2_1+x^2_2+\ldots+x^2_n}{n^2}$$ I have no idea how to approach this task. I was thinking that I could maybe $$ \frac{x^2_1+x^2_2+\ldots+x^2_n}{n^2}=\frac{\frac{x^2_1}{2}+\frac{x^2}{2}+\ldots+\frac{x^2_n}{2}}{n^2}+\frac{\frac{x^2_1}{2}+\frac{x^2}{2}+\ldots+\frac{x^2_n}{2}}{n^2}=\frac{\frac{x^2_1}{3}+\frac{x^2}{3}+\ldots+\frac{x^2_n}{3}}{n^2}+\ldots$$ But that doesn't lead anywhere. How should I approach it? What does the second limit ($\limsup\ldots\in\mathbb{R}$) mean and how should I use it?

Answer 1

I think it is enough to see: $$ \frac{x_1^2+\dots x_n^2}{n^2}\leq \frac{Max_{\{i=1,\dots,n\}}(x_i)}{n}\frac{x_1+\dots x_n}{n} $$

Now $$ \limsup_{n\to \infty} \frac{Max_{\{i=1,\dots,n\}}(x_i)}{n} =0, \quad \limsup_{n\to \infty} \frac{x_1+\dots x_n}{n}\in \mathbb R $$

Then $$ \limsup_{n\to \infty} \frac{x_1^2+\dots x_n^2}{n^2}=0. $$

as $\frac{x_1^2+\dots x_n^2}{n^2}>0$ we conclude $$ \lim_{n\to \infty}\frac{x_1^2+\dots x_n^2}{n^2}=0. $$