Take the 2-minute tour ×
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.

I am reading kreyszig functional analysis book where I got this problem:

Let $X$ be the set of all positive integers and $d(m,n) = \mid \frac{1}{m} - \frac{1}{n} \mid$. I have to show that $(X,d)$ is not complete metric space.

I took sequence $(x_n) = (n)$ which I showed that cauchy in $X$. But, I am not sure whether I am correct or not. I am also struggling with showing that this sequence is not convergent in $X$.

Thanks for helping me

share|improve this question
4  
What could it converge to? Use proof by contradiction –  Thomas Andrews Jun 1 '12 at 3:19
    
@JonasMeyer I will remember that. I have to edit now. –  srijan Jun 1 '12 at 3:54

1 Answer 1

up vote 1 down vote accepted

I think you are on the right track. You want to show that your sequence does not converge. If it did converge, it would have a limit $L$. So say "Suppose the sequence converges. Then it has a limit $L$ such that… (definition of limit)." Then prove a contradiction, which shows that no such limit $L$ can exist.

share|improve this answer
    
Suppose it has limit $L$ then $\lim _{n\rightarrow \infty}d(n, L) =\mid \frac{1}{L}\mid $. Now what argument should I give in support of our claim? –  srijan Jun 1 '12 at 3:35
2  
@srijan: That's good: For each positive integer $L$, $d(n,L)\to \frac{1}{L}$, and $\frac{1}{L}\neq 0$, so this shows that... –  Jonas Meyer Jun 1 '12 at 3:39
    
@JonasMeyer Thanks to both of you. –  srijan Jun 1 '12 at 3:53

Your Answer

 
discard

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.