I read a reference about the proof of “a bounded and monotone sequence is convergent”, in a certain PDF written by a physics professor.
Details of the reference is that the proposition “a bounded and monotone sequence is convergent” cannot be proved without the contradiction method.
But,I look up the proof of the theorem in other three books , This theorem is apparently not proved with indirect proof in any books.
I am very confused. Maybe, I think that we implicitly use reductio ad absurdum in the demonstration of the proposition.
I wish people familiar with this problem which answer.
That proposition can be proved without the contradiction method.
Proof: Assume that $(a_n)_{n\in\mathbb N}$ is increasing and bounded (the case in which it is decreasing is similar). Since the set $\{a_n\,|\,n\in\mathbb{N}\}$ is bounded above, it has a supremum $s$. I will prove that $\lim_{n\to\infty}a_n=s$. Take $\varepsilon>0$. Then, since $s-\varepsilon<s$, $s-\varepsilon$ is not an upper bound of $\{a_n\,|\,n\in\mathbb{N}\}$. So, there is some natural $N$ such that $a_n>s-\varepsilon$. And if $n\geqslant N$, then $s-\varepsilon<a_N\leqslant a_n\leqslant s$. In particular,$$n\geqslant N\implies\lvert a_n-s\rvert<\varepsilon.$$