Proving that the sequence $x_n = \frac{n+1}{3n-2}$ is Cauchy.
The definitions I've been given for Cauchy sequence are
1)$\forall \epsilon>0 ,\, \exists N \in\Bbb{N} \,$ $\ni $ $\forall n,m>N \ $ , it follows that $|x_m-x_n|<\epsilon$
2)$\forall \epsilon>0 ,\, \exists N \in\Bbb{N} \,$ $\ni $ $\forall n,>N \, p\in \Bbb{N} \ $ , it follows that $|x_n-x_{n+p}|<\epsilon$
So far I've worked out that for $m,n \in \Bbb{N}\,, |x_m-x_n| = |\frac{5(n-m)}{(3m-2)(3n-2)}|$, but I don't know how I can continue from there. Any suggestions?
Hint: $\frac{n+1}{3n-2}=\frac13+\frac53\cdot\frac{1}{3n-2}$