Countable family of Hilbert spaces is complete Let $H_1, H_2, \ldots, H_n$ be a countable family of Hilbert spaces. Let H be the set of tuples $x = (x_1, \ldots, x_n,\ldots)\in \prod_n H_n$
with the property that
$$\|x \| ^2 =\sum_n \| x_n \| _{H_n}^2 <\infty.$$ Show that H is a Hilbert space.
I have no problem showing that that it is a normed, but I have some problem with the completeness.
If we define $x^*$ as the coordinatewise limit points. How do we then show that $$\| x^* \| <\infty$$ and 
$$\lim_{k\rightarrow \infty} \| x^* - x_k \| \rightarrow 0 $$
 A: Let $(x_n)_{n\geq 1}$ a Cauchy sequence in $H$, then given $\epsilon>0$, there exists $N_o$ positive such that
\begin{equation}
\|x_n-x_m\|_H<\epsilon, \text{ for }n, m\geq N_o\qquad(*).
\end{equation}
from this equation we have, for each $k\geq 1$
$$\|x_n^k-x_m^k\|_{H_k}^2\leq\sum_{k}\|x_n^k-x_m^k\|_{H_k}^2=\|x_n-x_m\|^2<\epsilon^2;\text{ for } n, m\geq N_o.$$
This means that $(x_n^k)_{n\geq 1}$ is a cauchy secuence in $H_k$ for each $k\geq 1,$ so there exists $x_k\in H_k$ such that $x_n^k\to x_k$ as $n\to \infty.$ Now we will consider the element $x=(x_1,x_2,x_3,x_4,\ldots,x_n, \ldots )$ obtained in this way. It's not too much dificult to show that $x\in H$ and $x_n\to x$ as $n\to \infty.$ 
In order to show that $x\in H$, we will show that $x_{n_o}-x\in H$ for some $n_o,$ since $H$ is closed under subtraction it will follow that $x=x_{n_o}-(x_{n_o}-x)\in H.$ Pick $N\in \mathbb{N}$, by $(*)$
$$\sum_{k=1}^{N}\Arrowvert x_n^k-x_m^k\Arrowvert_{H_k}^2\leq \sum_{k}\Arrowvert x_n^k-x_m^k\Arrowvert_{H_k}^2=\Arrowvert x_n-x_m\Arrowvert_H^2<\epsilon^2, \text{ for } n,m\geq N_o.$$ 
Let $m\to \infty$ in the finite sum on the left hand side: we obtain 
$$\sum_{k=1}^{N}\Arrowvert x_n^k-x_k\Arrowvert_{H_k}^2<\epsilon^2, \text{ for } n\geq N_o,$$
 Since this equation holds for all $N\in \mathbb{N}$ we have, on letting $N\to \infty$,
 $$\sum_{k=1}^{\infty}\Arrowvert x_n^k-x_k\Arrowvert_{H_k}^2<\epsilon^2, \text{ for } n\geq N_o,$$
 that is
 $$\Arrowvert x_n-x\Arrowvert_H<\epsilon, \text{ for } n\geq N_o, \qquad(**)$$
 In particular we have 
 $$\Arrowvert x_{N_o}-x\Arrowvert_H<\epsilon,$$
 and so $x_{N_o}-x\in H, $ this proves that $x\in H,$ at the same time from $(**)$ we get $x_n\to x$ as $n\to\infty$ with recpect the norm $\Arrowvert\cdot \Arrowvert_H$.$\clubsuit$
