I'm trying to prove that the sequence $\left(\frac{1}{2},\frac{1}{3},\frac{2}{3},\frac{1}{4},\frac{2}{4},\frac{3}{4},\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5},\cdots\right)$ is not a Cauchy sequence.
I know that a sequence of real numbers is not Cauchy if there exists an $\epsilon>0$ such that, for all $N\in\mathbb{N}$, there exists $m,n>N$ such that $|x_{m}-x_{n}|\geq\epsilon$. It intuitively makes sense to me that the sequence cannot be Cauchy, as the distance between points where the denominator changes (like $\cdots\frac{99}{101},\frac{100}{101},\frac{1}{102},\cdots$) keeps growing larger. However, I'm not sure how to find indices $m$ and $n$ in general with $|x_{m}-x_{n}|\geq\epsilon$. Thanks in advance for any help!