The alternating harmonic sums converge

Monotonicity.
$$S_{2n+2}-S_{2n}=\frac1{2n+1}-\frac1{2n+2}>0,\qquad S_{2n+3}-S_{2n+1}=-\frac1{2n+2}+\frac1{2n+3}<0.$$
So $(S_{2n})$ is increasing and $(S_{2n+1})$ is decreasing.

Difference. $S_{2n+1}-S_{2n}=\frac1{2n+1}\to0$.

Adjacent sequences. They converge to the same limit $\ell$. Since both the even and the odd subsequences tend to $\ell$, the whole sequence $(S_n)$ converges to $\ell$.

Remark. The limit is $\ln2\approx0.6931$ ($S_{2000}\approx0.6929$). The bounds $S_{2n}\le\ln2\le S_{2n+1}$ give an error estimate of $\frac1{2n+1}$ for free.