Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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


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

share|cite|improve this question
When is $d(m,n)<1/2$? – David Mitra Jan 29 '14 at 21:53
Please show your working so far :) – Shauna Jan 29 '14 at 22:04

1 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 $\Bbb R$.
share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.