Monotone Convergence Theorem (Real Analysis)

Theorem
Every bounded monotone sequence is convergent.

Let $\left \langle {x_n} \right \rangle$ be a sequence in $\R$.

Proof
Let $$\{ a_n \}$$ be such a sequence. By assumption, $$\{ a_n \}$$ is non-empty and bounded above. By the least-upper-bound property of real numbers, $$c = \sup_n \{a_n\}$$ exists and is finite. Now, for every $$\varepsilon > 0$$, there exists $$N$$ such that $$a_N > c - \varepsilon $$, since otherwise $$c - \varepsilon $$ is an upper bound of $$\{ a_n \}$$, which contradicts to the definition of $$c$$. Then since $$\{ a_n \}$$ is increasing, and $$c$$ is its upper bound, for every $$n > N$$, we have $$|c - a_n| \leq |c - a_N| < \varepsilon $$. Hence, by definition, the limit of $$\{ a_n \}$$ is $$\sup_n \{a_n\}.$$