If we have $\sum (-1)^nx_n$ and if $x_n>0$, and $\lim_{n\rightarrow +\infty }x_n=0$ and $x_n$ is a decreasing sequence then
$\left | \sum_{n=q+1}^{p} (-1)^nx_n\right |\leq x_{q+1}$
And if $\underset{x}{sup}(x_{q+1})\rightarrow0 $ then the series converges uniformly
Presumably $x_n$ is a function of $x$ in some domain $D$ as you are concerned with uniform convergence.
Since $x_n$ is a decreasing sequence, $x_{n} - x_{n+1} > 0$ and
$$\left | \sum_{n=q+1}^{p} (-1)^nx_n\right| = \left |(-1)^{q+1}(x_{q+1} - x_{q+2} + x_{q+3} - x_{q+4} + \ldots) \right| \\= x_{q+1} - (\, x_{q+2} - x_{q+3}\, ) - (\, x_{q+3} - x_{q+4}\, ) - \ldots \leqslant x_{q+1} \leqslant \sup_{x \in D} x_{q+1} $$
Thus, if the RHS tends to $0$ as $q \to \infty$ then uniform convergence follows as the uniform Cauchy criterion is satisfied.