Convergence of partial sums of an alternating series

Convergence of partial sums of an alternating series

Suppose $x_0, x_1, x_2,\ldots$ is a sequence of positive numbers
monotonically converging to zero. Then $x_o - x_1 + x_2 -x_3+\cdots$
converges.
How would you prove this statement? I know you would have to either prove
that the sequence of partial sums has an upper bound or consider $n, m >
N$ and consider the series between $x_n$ and $x_m$ and prove it is less
than any real epsilon, but not exactly sure how to carry the proof
rigorously out.