Cantor's proof that every bounded monotone sequence of real numbers converges Cantor constructed the field of real numbers by using Cauchy sequences.
According to him every Cauchy sequence of real numbers converges (correct me if I'm wrong).
So how did he prove that every bounded monotone sequence of real numbers converges by using this property?
 A: Take as given that every Cauchy sequence of real numbers converges. Suppose $a_n$ is a sequence of real numbers which is monotone and not convergent. Since it is not convergent, it is not Cauchy, so 
$$(\exists \varepsilon > 0)(\forall N \in \mathbb{N})(\exists m,n \in \mathbb{N}) \: m>n\geq N \text{ and }  |a_m - a_n| > \varepsilon.$$
Fix such a $\varepsilon$, then use this statement inductively to get a subsequence $a_{n_k}$ with $|a_{n_{k+1}} - a_{n_k}|>\varepsilon$ for every $k$. To see how to do this, start with $N=1$, get $m,n$ and call them $n_1$ and $n_0$. Now use $N=n_0$, get $m,n$, one will be the same $n_1$ and you will also get an $n_2$. Repeat this procedure.
Now by the monotonicity, $a_{n_{k+1}} > a_{n_k} + \varepsilon$ and so $a_{n_k} > a_{n_0} + k \varepsilon$. Hence $a_{n_k}$ is not bounded so $a_n$ is not bounded either. We have shown that if a sequence of real numbers is monotone and not convergent then it is not bounded. The desired statement follows by contraposition.
