Is sum of two square summable sequences is again square summable?

Question

I am trying to prove the set $W=\{(a_n) | \sum a_n^2 <∞\}$ is subspace of vector space $V=\mathbb{R}^∞$ (vector space of all sequences of real numbers)

For this, clearly as $0^2+ 0^2+...=0 <∞$ Hence (0) is in $W$.

Further, if $(a_n), (b_n) \in W$ then we have $\sum a_n^2<∞$ and $\sum b_n^2<∞$.

How can I prove $\sum (a_n+b_n)^2<∞$

Answer 1

Hint By Cauchy-Schwarz: $$(a_n+b_n)^2 \leq 2 (a_n^2+b_n^2)$$