Describe all the convergent and Cauchy sequences in this metric space

Question

Consider the set of natural numbers $\mathbb N$ with the metric

$$d(m,n)=\frac{\left|m-n\right|}{1+\left|m-n\right|}$$

Describe all convergent sequences and all Cauchy sequences in this metric space. Is the metric space $(\mathbb N, d)$ complete?

Answer 1

Hints: We have

• $a_n\to a$ in the metric space $(\Bbb N,d)$ iff $d(a_n,a)\to 0$ in $\Bbb R$.
• For a sequence $c_n\in\Bbb R$ we have $\displaystyle\frac{c_n}{1+c_n}=1-\frac1{1+c_n}$ and its limit is $0$ iff $\displaystyle\frac1{1+c_n}\to 1\ $ iff $\ (1+c_n)\,\to 1\ $ iff $\ c_n\to 0$.
• Similar considerations can lead to that Cauchy sequences are also Cauchy in $\Bbb R$.