Show that the sequence is Cauchy.

Question

I need to show that $x_n=\frac{1}{n}(1+\frac{1}{4}+\cdots+\frac{1}{3n-2})$ is a Cauchy sequence.

For $n \leq m$ , $|x_m-x_n|\leq|\frac{1}{n}(\frac{1}{3n+1}+\frac{1}{3n+4}+\cdots+\frac{1}{3m-2})|$ How should I proceed further?

Answer 1

The sequence $(\frac{1}{3n-2})$ is convergent. So by Cauchy's first theorem on limits the sequence $\frac{1}{n}(1+\frac{1}{4}+\dots+\frac{1}{3n-2})$ is also convergent. Since every convergent sequence is Cauchy so this sequence is Cauchy as well.