I am trying to understand better Cauchy sequence in the book that I am reading he is trying to provide counter argument that in general metric spaces we can find sequence that are Cauchy but aren't convergent for example $\frac{1}{n}$ is Cauchy but doesn't converge in the space T = (0,1].
I am trying to prove that the sequence is Cauchy to begin with if we consider $n \geq m > 0$ Consider $|a_n - a_m| = \frac{m - n}{mn} < \frac{m}{mn} = 1/n < \epsilon$. So if we choose N = $\frac{1}{\epsilon}$ that should work since for $n, m > N = \frac{1}{\epsilon}$
we have the following:
$|a_n - a_m| = \frac{m - n}{mn} < \frac{m}{mn} = \frac{1}{n} < \epsilon$.
what do you guys think any the proof is good or is there anything to make it better ? It is always good to learn from the extra bit of details as it makes my brain understand the subject more !
Take an arbitrary $\epsilon > 0$. I think you can say that \begin{equation*} |a_n - a_m| = \left| \frac{1}{n} - \frac{1}{m} \right| \leqslant \frac{1}{n} + \frac{1}{m}. \end{equation*} If you choose an index $N \in \mathbb{N}$ greater than $\dfrac{2}{\epsilon}$, then you have \begin{equation*} |a_n - a_m| \leqslant \frac{1}{n} + \frac{1}{m} \leqslant \frac{1}{N} + \frac{1}{N} < \epsilon \quad \text{for all}\ m, n \geqslant N. \end{equation*} Therefore, $\left\{\dfrac{1}{n}\right\}$ is a Cauchy sequence.