why $N = 1 / \epsilon$?

Question

Here is the part of the text I am asking about:

I am wondering, why in the proof that the sequence $x_m$ is Cauchy the writer used that $N = 1/ \epsilon,$ could someone explain this to me please?

Answer 1

The area of the triangle pictured in the second figure is $\frac12\left\lvert\frac1m - \frac1n\right\rvert \le \frac12\left(\frac1m + \frac1n\right)$ (by triangle inequality). Thus, if you pick both $m$ and $n$ greater than $\frac 1\varepsilon$, you will have the area is $< \varepsilon$.

Answer 2

If you go far enough out in the sequence that $m, n \geq N > \frac{1}{\varepsilon}$, then you have $$ \frac{1}{m}, \frac{1}{n} \leq \frac{1}{N} < \varepsilon, $$ and the area of that triangle is less than $$ \frac{1}{2m}, $$ assuming WLOG that $m < n$ (if not just swap the roles).

Does this help you understand the proof?