$S=\{(\lambda_1,\lambda_2,...)|\sum_v \lambda_v^2<\infty\}$.Then$\sum_v \lambda_v \eta_v$ converges & $(x,y) =\sum_v \lambda_v\eta_v$ is inner product

Question

In the book of linear algebra by Werner Greub, at page 191, it is asked that

Consider the space $S$ of all infinite sequences $x = (\lambda_1, > \lambda_2, ...)$, where $\lambda_v \in \mathbb{R}$, such that $$\sum_v \lambda_v^2 < \infty.$$ Show that $\sum_v \lambda_v \eta_v$ converges and that the bilinear function $(x,y) =\sum_v \lambda_v \eta_v$ is an inner product in $S$.

So how can we show this ?

I mean since the sum $\sum_v \lambda_v^2$ is finite, $\lim_{v\to \infty} \lambda_v^2$ = 0, hence $\lim_{v\to \infty} \lambda_v$ = 0.

Moreover, we also know that $\frac{1}{v} > \lambda_v^2$, so $\sqrt{\frac{1}{v}} > \lambda_v$, hence $\frac{1}{v} > \lambda_v \eta_V$, but this doesn't give me anything useful.

Answer 1

Just a hint.

Remember that $2ab \leq a^2 + b^2$ for every pair of real numbers.