Let $H_1, H_2, \ldots $ be a countable set of Hilbert spaces. Let $H\subset \prod_k H_k$ be the set where
$\|x\|^2 = \sum_k \|x_k\|^2_{H_k} < \infty.$
Show that $H$ is a Hilbert space.
It is very easy to see that a Cauchy sequence in $H$ converges element-wise to an element $x^*$. But how would one show that this element is in $H$, i.e. that $\|x\|^2 < \infty$?