Prove that $a:\mathbb{Z}_+\to \mathbb{R}$, with $a_n = \frac{1}{n}$ is a Cauchy sequence.

Question

I am not allow to use the fact if a sequence is convergent then its Cauchy which is the only way I can solve this problem.

Answer 1

Just use the definition of Cauchy sequence.

Let $\epsilon>0$ and show that there is $N\geq 1$ (which depends on $\epsilon$) such that for all $n,m\geq N$, $\left|\frac{1}{n}-\frac{1}{m}\right|<\epsilon$.

Hint. Note that $\left|\frac{1}{n}-\frac{1}{m}\right|\leq \frac{1}{\min(m,n)}$,