I am trying to prove:
If a sequence is monotone and bounded then it converges.
My idea is: Assume $a_n$ is monotone and not converges and then show that it is not bounded. But: my problem is that I fail to prove it is not bounded. Please can you help me?
I found a different proof in a book but I want to know if my proof can work.
Best Answer
Without loss of generality assume that $(a_n)$ is increasing and bounded above (the other case is similar) then the set $$A=\{a_n\;|\; n\in\mathbb N\}$$ has a supremum $s=\sup A$ and we know by the characterization of this supremum: $$\forall \epsilon>0,\; \exists a_p\in A \;|\; s-\epsilon\le a_p\le s$$ but since $(a_n)$ is increasing then $$\forall \epsilon>0,\; \exists p\in\mathbb N, \forall n\ge p \;|\; s-\epsilon\le a_p\le a_n\le s$$ which means that $\displaystyle\lim_{n\to\infty}a_n=s$.