Proving that $x,y \in \ell^2(\Bbb N) \implies x+y \in \ell^2(\Bbb N)$.

Question

I want to prove that $x,y \in \ell^2(\Bbb N) \implies x+y \in \ell^2(\Bbb N)$. I'm damn sure that there is a quick way to do this, but I'm not seeing it. I am capable of proving Young, Hölder and Minkowski's inequalities to estabilish the result for $\ell^p(\Bbb N)$, but that seems overkill here and I don't want to do that.

Can someone point me the way, please? Thanks.

Obs.: $\ell^2(\Bbb N) = \left\{ (x_n)_{n \in \Bbb N} \mid x_n \in \Bbb C~ \forall\,n, \text{ and } \sum_{n \in \Bbb N}|x_n|^2 < +\infty \right\}$

Answer 1

Absolutely cloddish inequality: If $a,b \ge 0,$ then $(a+b)^2 \le 4a^2 + 4 b^2.$ Proof: If $a\le b,$ then the left side is $\le (2b)^2 = 4b^2,$ same idea of course if $a\ge b.$ So $$\sum (x_n+y_n)^2 \le \sum (|x_n|+|y_n|)^2 \le \sum (4|x_n|^2 + 4|y_n|^2)$$ and that does it.

Answer 2

Hint: multiply out the terms in the sum for $||x + y||$, and use Cauchy-Schwarz to find a bound for $\sum x_iy_i$ in terms of $||x||$ and $||y||$.

Answer 3

This follows from the triangle inequality of the$\ell^2$ norm, since $x,y \in \ell^2$ then $\|x\|_2 <\infty $ and $\|y\|_2 <\infty$ thus $$ \sum_{n=0}^\infty |x_n + y_n|^2 =\| x+ y\|_2^2 \leq (\|x\|_2+\|y\|_2)^2 < \infty $$ Hence, indeed $x+y \in \ell^2$