6
$\begingroup$

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.

  • 0
    @Henry: That was a clever fix, I was thinking of $n \pmod {2}$. Thanks.2012-09-30

2 Answers 2

7

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?
  • 0
    Great hint! Thanks a lot ;)2012-09-30
3

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 $H_{n}-H_{n-1}=\frac{1}{n} \to 0$.

  • 0
    (But of course that's easy to fix. The asker is obviously just confused about the notion of a non-decreasing sequence)2012-09-30