Proving Cauchy sequence and its convergence

Question

As I am having trouble in concluding the question, please look at this also: For a sequence $$1 - \frac{1}{2} +\frac{1}{3}+\cdots +\frac{(−1)^{n−1}}{n} \\ \text{Then}\\ |s_n−s_m| = \left|\frac{(−1)^{m−1}}{m} +\frac{(−1)^{m}}{m+1} +\cdots+\frac{(−1)^{n−1}}{n}\right| \\ \leq | \sum_{k=m}^{n} \frac{(−1)^{k−1}}{k} | \\ \leq \sum_{k=1}^{\infty} \frac{(−1)^{k−1}}{k} $$ Since the sequence is monotonically increasing $$\leq\log2,$$ so can we end the question by saying: For each $\epsilon\geq\log2≥0$, there exists a positive integer $M\geq\epsilon$ such that for $n,m\geq M$, we have that $|s_n−s_m|\leq\epsilon$, and therefore it is convergent. But I came to know that my proof does not work for any epsilon i e. Less then log2 , please help me to correct it.