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.

Given a non-decreasing sequence $(a_n)$: $$a_1 \leq a_2 \leq a_3 \leq a_4 \ldots$$ and $$\displaystyle\lim_{n\to\infty}(a_n - a_{n-1}) = 0$$ Does it have to converge?
For strictly than sequence $a_1 < a_2 < a_3 < a_4 < \ldots$ with the limit property, it's easy to show that it doesn't converge, for example take $a_n = \sqrt{n}$. In this case, however I couldn't find a counter example sequence, and I have a feeling this sequence might converge but again I'm not so sure. Any hint would be greatly appreciated.

share|improve this question
    
$a_n = \sqrt{n}$ also satisfies $a_1 \leq a_2 \leq a_3 \leq a_4 \ldots$. So too does $a_n = \sqrt{\lfloor n/2\rfloor}$ –  Henry Sep 30 '12 at 9:02
    
As Henry indicates, you seem to have answered this yourself already. A strictly increasing sequence is in particular a non-decreasing sequence, so. . . –  saurs Sep 30 '12 at 13:00
    
@Henry: That was a clever fix, I was thinking of $n \pmod {2}$. Thanks. –  Chan Sep 30 '12 at 19:05

2 Answers 2

up vote 7 down vote accepted

Clearly the the sequence $b_n=a_{n+1}-a_n$ is non-negative, i.e. $b_n\ge0$ for each $n$.

  • If any non-negative sequence $b_n\ge0$ is given, can you find a corresponding (non-decreasing) sequence $a_n$ such that $b_n=a_{n+1}-a_n$?
  • Can you find a non-negative sequence which does not have limit?
share|improve this answer
    
Great hint! Thanks a lot ;) –  Chan Sep 30 '12 at 6:58

Take for example the harmonic sequence: $$ H_n = 1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}$$

It has the property that $H_n \to \infty$, but $H_{n}-H_{n-1}=\frac{1}{n} \to 0$.

share|improve this answer
    
For some reason, the asker seems to want an example of a non-decreasing but non-strictly increasing sequence, so this wouldn't meet the criteria. –  saurs Sep 30 '12 at 13:07
    
(But of course that's easy to fix. The asker is obviously just confused about the notion of a non-decreasing sequence) –  saurs Sep 30 '12 at 15:16

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.