Prove that for a sequence in $\Bbb N$, (as $j \rightarrow \infty$, $n_j \rightarrow \infty$) $\Rightarrow$ Cauchy sequence

Question

Given a metric space $(\Bbb N,d)$ where $d(x,y)= |\frac{1}{x}-\frac{1}{y}|.$

I need to prove that for a sequence $(n_j)_{j \in \Bbb N} \in \Bbb N$, (as $j \rightarrow \infty$, $n_j \rightarrow \infty$) $\Rightarrow$ this sequence is a Cauchy sequence.

Here is the definition of Cauchy seuence:

Let $(x_n)^∞ _{n=1} ⊂ X$ where $(X, d)$ is a metric space. Then $(x_n)$ is a Cauchy sequence if for every $\varepsilon > 0$ there exists an integer $N$ such that $m, n ≥ N ⇒ d(x_m, x_n) < \varepsilon$ .

By definition, I need an explicit $N$ for every $\varepsilon >0$ to prove that. I have spent some time on this question but did not make much progress. Can someone help me, please?

Answer 1

Take $\epsilon > 0$. Take $M$ large so that $\frac{1}{M} < \epsilon$. Take $N$ large so that $\forall j \ge N, n_j \ge M$ which is possible since $n_j \to \infty$. Then, for $j,k \ge N$, $d(n_j,n_k) = |\frac{1}{n_j}-\frac{1}{n_k}| \le \max(\frac{1}{n_j},\frac{1}{n_k}) \le \frac{1}{M} < \epsilon$.