had this difficult question from a textbook, and I haven't been able to figure out the solution.

say we have a sequence of bounded real numbers $a_n$ such that $2a_n \leq a_{n-1} + a_{n+1} \forall n\in\mathbb{N}$. Show that this sequence converges.

What ive tried: I did some algebra on it then tried to use the cauchy criterion, since we don't know the limit, but we know the relation between terms.

also tried moving things around and reindexing and using boundedness. but havent been able to come up with anything there either.

since bounded and monotone converges.

any help would be appreciated

  • $\begingroup$ Getting uppercase and lowercase letters right should be rather easy. $\endgroup$ – TobiMcNamobi May 10 '16 at 8:30
  • $\begingroup$ i dont know what you mean by that. $\endgroup$ – mac5 May 10 '16 at 17:56
  • 1
    $\begingroup$ :-) Maybe I Underestimated The Problem. $\endgroup$ – TobiMcNamobi May 11 '16 at 6:51

Define $b_n=a_{n+1}-a_n$. Then $b_n$ is bounded and the given condition becomes $$b_{n-1}\le b_n.$$ So $b_n$ is monotonically increasing and hence it converges.

Now we have $$a_n=a_0+\sum_{i=0}^{n-1}b_i$$

Let $$\lim_{n\to\infty}b_n=c > 0$$.

Then there exists $N$ such that $b_n>c/2$ for all $n>N$, and hence $a_n$ will not be bounded for sufficiently large $n$. The case is similar for $c < 0$. So we conclude that $c=0$.

Because $b_n$ is increasing and the limit is $0$, $b_n \le 0$.

Hence $a_n$ is decreasing. Because $a_n$ is bounded, so it converges.

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