Monotone Convergence Theorem (for sequences)

Monotone Convergence Theorem (sequences): If $(x_n)$ is a monotone increasing sequence in $\mathbb{R}$ that is bounded above , then $(x_n)$ converges and $\lim_{n\to\infty} x_n = \sup\{x_n:n\in\mathbb{N}\}.$ Similarly, a monotone decreasing sequence that is bounded below converges to its infimum .

This theorem is one of the main working consequences of completeness and provides a robust method for proving convergence without explicitly identifying a limit.

Proof sketch (optional): Let $s=\sup\{x_n\}$. Given $\varepsilon>0$, $s-\varepsilon$ is not an upper bound, so some $N$ has $x_N>s-\varepsilon$. Monotonicity gives $s-\varepsilon<x_n\le s$ for all $n\ge N$, proving $x_n\to s$.