I have the following problem. Prove by definition that the sequence
$r_{n}=\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+...+\frac{(-1)^{n}}{2n+1}$
Is a Cauchy sequence.
I think I almost have it, but I have not been able to conclude anything of a series of inequalities that I obtained and I would like you to give me some advice or if I have an error or what I lack to conclude.
What I have is:
Sup. $n>m$
$|r_{n}-r_{m}|=|\sum_{k=m+1}^{n} \frac{(-1)^{k}}{2k+1}|\leq\sum_{k=m+1}^{n} \frac{1}{2k+1}<\sum_{k=m+1}^{n} \frac{1}{k}$
But from there I don't know what else to conclude