[Math] $\sum x_n^2$ and $\sum y_n^2$ converge, what does $\sum (x_n + y_n)^2$ do

Question

So I know that $\sum x_n^2$ converges, and $\sum y_n^2$ converges. ${x_n}$ and ${y_n}$ are both sequences of real numbers. How can I prove that $$\sum(x_n+y_n)^2$$ converges? So far I have
$$\sum(x_n+y_n)^2=\sum(x_n)^2 +\sum (y_n)^2 +\sum(2x_ny_n)$$
so I need to prove that $\sum 2x_ny_n$ converges. Is this the right way to approach it? If it is, I am completely lost as to the next step.

I also need to prove that $$\sqrt{\sum(x_n+y_n)^2}\leq \sqrt{\sum(x_n)^2}+\sqrt{\sum(y_n)^2}$$

I assume this will follow pretty easily from a result in the first proof, but I'm stuck on that now, and perhaps it is easier to start with the second one.
I have a lot of work I've done on the first problem, but none of it seemed to go anywhere. Any hints?

Thanks!

Answer 1

$$x^2 + 2xy + y^2 \geq 0$$

Similarly,

$$ x^2 - 2xy + y^2 \geq 0$$

Therefore

$$ -x^2 - y^2 \leq 2xy \leq x^2 + y^2 $$

So

$$ |2xy| \leq x^2 + y^2 $$

This shows that $s_n = \displaystyle\sum_{i=1}^n |2x_i y_i|$ is bounded and increasing sequence and so is convergent. So the series $\displaystyle\sum_i 2x_iy_i$ is absolutely convergent and therefore must be convergent.