Partial sums of alternating Harmonic series satisfy Cauchy Criterion?

Hello

I am a little stuck in seeing how the partial sums of an alternating Harmonic series satisfy the cauchy condition.

i.e. The partial sums are defined as $\displaystyle s_n=\underset{i=0}{\overset{n}{\sum }}\frac{(-1)^i}{i}$

This yields for $\displaystyle n<m$

$\displaystyle \left|s_m-s_n\right|=\left|\frac{1}{n+1}-\frac{1}{n+2}+\text{...} \pm \frac{1}{m}\right|<\frac{1}{n}$

Its this last inequality I am having trouble verifying. Any ideas?