It is known in France as the 'theorème spécial pour les séries altérnées', and it is part of any course on series... Pierre.

In english, it's the alternating series test. The idea is that we can put $\sum^{n}_{i=0} (-1)^i=A_n$, and then write $\left|\sum^q_{i=p} (-1)^ix_i\right|=\left|\sum^{q-1}_{i=p} A_i(x_i-x_{i+1})-A_qx_q-A_{p-1}x_p\right|$ $A_i\leq 1$ for each $i$, so we have $\left|\sum^{q-1}_{i=p} A_i(x_i-x-{i+1})-A_qx_q-A_{p-1}x_p\right|\leq \left|\sum^{q-1}_{i=p} (x_i-x_{i+1})-x_q-x_p\right|=2x_p$ But as $p\to\infty$, $x_p\to 0$, so we have convergence from the Cauchy criterion (that is, Cauchy sequences converge in $\mathbb{R}$).

$A_{2n}$ decreasing and $A_{2n+1}$ increasing $A_{2n+1}-A_{2n}=-s_{2n+1}$ $\Rightarrow $ QED

Why do I think the only sequence is Xn=0 since Xn is decreasing?