Let $(s_n)_{n \in \mathbb{N}}\in\ell^2(\mathbb{N})$ (i.e. $\displaystyle \sum_{n=0}^{\infty}\vert s_n\vert^2<\infty$). Define vectors $A=[A_1,\ldots,A_M]$ and $B=[B_1,\ldots,B_M]$ with coordinates $A_k=\displaystyle\sum_{i=k}^{M}|s_i|$ and $B_k=\displaystyle\sum_{i=M}^{2M-k}|s_i|,\ k=1,\ldots,M\ $ (note that the length of the vectors depends on $M$).
Is it true that
$\frac{1}{\sqrt{M}}\Vert A-B\Vert_2\to0\ \text{as}\ M\to \infty ?$
Sorry, I'm an economist and not a mathematician so my symbols might be off. Experimental data and model simulations suggest the above to be true with changing $M$ (i.e., error is less than $1/\sqrt{M}$ for all $M$, hence $1/\sqrt{\infty}=0$ at $M=\infty$), but can it be shown analytically?
I reason:
$\frac{1}{\sqrt{M}}\Vert A_k-B_k\Vert_2=\frac{1}{\sqrt{M}}\left(\sum_{(i=something)}\Vert s_i\Vert_2-\sum_{(i=something else)}\Vert s_i\Vert_2\right)\to0$
hence $\frac{1}{\sqrt{M}}\Vert A-B\Vert_2\to0$.
But I think this is wrong because $\frac{1}{\sqrt{M}}\Vert A-B\Vert_2=M \frac{1}{\sqrt{M}}\Vert A_k-B_k\Vert_2\nrightarrow 0$. What's wrong with my reasoning?