We know that if $(x_n)_{n\in\mathbb{N}}$ is a sequence of element in a Hilbert space $(\mathcal{H}, \langle\cdot,\cdot\rangle)$ such that for $n\neq m$, $x_n\perp x_m$. Then the series $\sum_{n\in\mathbb{N}} x_n$ converges if and only if the series $\sum_{n\in\mathbb{N}}\Vert x_n\Vert^2$ converges, and in that case, we have $$ \Bigl\Vert \sum_{n\in\mathbb{N}}x_n\Bigr\Vert^2 = \sum_{n\in\mathbb{N}}\Vert x_n\Vert^2 $$
Now, if we take another sequence $(x_n)_{n\in\mathbb{N}}$ in $\mathcal{H}$ such that there exists $N_0\geqslant 1$ such that if $\vert n-m\vert\geqslant N_0$, then $x_n\perp x_m$. We also suppose that $\sum_{n\in\mathbb{N}}\Vert x_n\Vert^2$ exists (i.e. is finite). Prove that the series $\sum_{n\in\mathbb{N}}x_n$ is convergent and that there exists a constant $C$ which only depends on $N_0$, such that $$ \Bigl\Vert \sum_{n\in\mathbb{N}}x_n\Bigr\Vert^2 \leqslant C\sum_{n\in\mathbb{N}}\Vert x_n\Vert^2 $$
My try: Like in the Parseval equality proof, I try to prove that $S_q=\sum_{n\leqslant q}x_n$ is a Cauchy sequence: $$ \begin{align} \Vert S_{q+q'}-S_q\Vert^2 &=\sum_{n=q+1}^{q'}\Vert x_n\Vert^2+\sum_{n=q+1}^{q'}\sum_{j=q+1}^{q'}\langle x_n,x_j\rangle + \overline{\langle x_n,x_j\rangle}\\ &=\sum_{n=q+1}^{q'}\Vert x_n\Vert^2+\sum_{n=q+1}^{q'}\sum_{j\in I_n}\langle x_n,x_j\rangle + \overline{\langle x_n,x_j\rangle}\\ &\leqslant \sum_{n=q+1}^{+\infty}\Vert x_n\Vert^2+2\sum_{n=q+1}^{q'}\sum_{j\in I_n}\vert\langle x_n,x_j\rangle\vert\\ &\leqslant \sum_{n=q+1}^{+\infty}\Vert x_n\Vert^2+2\sum_{n=q+1}^{q'}\Vert x_n\Vert\sum_{j\in I_n}\Vert x_j\Vert \end{align} $$ where $I_n=\{j:\vert n-j\vert\leqslant N_0\}$. I know that $\vert I_n\vert\leqslant N_0$, but I can only seem to have $$ \sum_{j\in I_n}\Vert x_j\Vert\leqslant N_0\sup_{j\in I_n}\Vert x_j\Vert $$
I cannot seem to go further than that. I am able to bound $\sup_{j\in I_n}\Vert x_j\Vert$ by $C_j\Vert x_q\Vert^2$ if $x_q\neq 0$ with $C_j\in\mathbb{R}$ but this does not seem to be sharp enough, as the sequence $(C_j)_{j\in\mathbb{N}}$ is not easily bounded.
The same happens if I consider $\Vert S_q\Vert$ when trying to prove the inequality. Note that $N_0=1$ is the same case as in the Parseval equality.
Any help would be appreciated, thanks!
In fact, you have nearly worked out it. If we choose $q=0$ and $m=q+q'=q'$ in your step, We will have $$\Vert S_{m}\Vert^2 \leqslant \sum_{n=1}^{+\infty}\Vert x_n\Vert^2+2\sum_{n=1}^{m}\Vert x_n\Vert\sum_{j\in I_n}\Vert x_j\Vert$$ And we can estimate the second term: $$\begin{align} 2\sum_{n=1}^{m}\Vert x_n\Vert\sum_{j\in I_n}\Vert x_j\Vert&=\sum_{n=1}^{m}\sum_{j\in I_n}2\Vert x_j\Vert\Vert x_n\Vert\\&\le \sum_{n=1}^{m}\sum_{j\in I_n}(\Vert x_j\Vert^2+\Vert x_n\Vert^2)\\&=\sum_{n=1}^{m}\sum_{n-N_0\le j\le n+N_0}(\Vert x_j\Vert^2+\Vert x_n\Vert^2), \end{align}$$ here $x_j=0$ if $j\le 0$. and we have $$\begin{align} &\sum_{n=1}^{m}\sum_{n-N_0\le j\le n+N_0}(\Vert x_j\Vert^2+\Vert x_n\Vert^2)\\ &\le \sum_{n=1}^{m}(2N_0 \Vert x_n\Vert^2+\sum_{n-N_0\le j\le n+N_0}\Vert x_j\Vert^2)\\ &=2N_0\sum_{n=1}^{m}\Vert x_n\Vert^2+\sum_{n=1}^{m}\sum_{n-N_0\le j\le n+N_0}\Vert x_j\Vert^2\\ &=2N_0\sum_{n=1}^{m}\Vert x_n\Vert^2+\sum_{n=1}^{m}\sum_{|n-j|\le N_0}\Vert x_j\Vert^2\\ &\le 2N_0\sum_{n=1}^{\infty}\Vert x_n\Vert^2+\sum_{n=1}^{\infty}\sum_{j=1}^{\infty}\Vert x_j\Vert^2 I_{\{|n-j|\le N_0\}}\\ &=2N_0\sum_{n=1}^{\infty}\Vert x_n\Vert^2+\sum_{j=1}^{\infty}\sum_{n=1}^{\infty}\Vert x_j\Vert^2 I_{\{|n-j|\le N_0\}} \\ &\le 2N_0\sum_{n=1}^{\infty}\Vert x_n\Vert^2+\sum_{j=1}^{\infty}2N_0\Vert x_j\Vert^2 \\&\le 4N_0\sum_{n=1}^{\infty}||x_n||^2 \end{align} $$ Where $I_{\{|n-j|\le N_0\}}=1$ if $|n-j|\le N_0$,and $0$ otherwise.So we have $$||S_m||^2\le (4N_0+1)\sum_{n=1}^{\infty}||x_n||^2$$ for all $m>0$. Hence $C=4N_0+1$ is the desired constant.