Use Cauchy's convergence criterion to prove convergence of $x_n$
$x_n=1-\frac{1}{2}+\frac{1}{3}-\cdots+(-1)^{n+1}\frac{1}{n}$
as far as I am concerned,supposing that $m>n$, it's apparently warranted that $|x_m-x_n|=\sum_{k=n+1}^m (-1)^{k+1}\dfrac{1}{k}$
but, furthermore, it was showed on the textbook that $\sum_{k=n+1}^m (-1)^{k+1}\dfrac{1}{k}<\dfrac{1}{n+1}$
I wonder how can we approach the last step