Monotone Convergence Theorem

The Monotone Convergence Theorem (MCT) is one of the most fundamental results in real analysis. It states that every bounded monotone sequence in $\mathbb{R}$ converges. This theorem is a direct consequence of the completeness axiom and is the foundation for integration theory, measure theory, and countless limit arguments.

Statement

Theorem2.1Monotone Convergence Theorem

Let $(a_n)$ be a sequence in $\mathbb{R}$ .

If $(a_n)$ is increasing (i.e., $a_n \leq a_{n+1}$ for all $n$ ) and bounded above, then $(a_n)$ converges to $\sup\{a_n \mid n \in \mathbb{N}\}$ .
If $(a_n)$ is decreasing and bounded below, then $(a_n)$ converges to $\inf\{a_n \mid n \in \mathbb{N}\}$ .

RemarkIntuition

An increasing sequence that is bounded above cannot "escape to infinity" — by completeness, it must approach a limit, which is its supremum. The sequence "climbs up" toward the least upper bound.

Proof

We prove (1); part (2) follows by considering $(-a_n)$ .

Let $(a_n)$ be increasing and bounded above. By the completeness axiom, $S = \{a_n \mid n \in \mathbb{N}\}$ has a supremum. Let $L = \sup S$ .

We show $a_n \to L$ . Given $\epsilon > 0$ , since $L$ is the least upper bound of $S$ , there exists $N \in \mathbb{N}$ such that

$a_N > L - \epsilon$

(otherwise $L - \epsilon$ would be a smaller upper bound). Since $(a_n)$ is increasing, for all $n \geq N$ ,

$L - \epsilon < a_N \leq a_n \leq L < L + \epsilon.$

Thus $|a_n - L| < \epsilon$ for all $n \geq N$ , so $a_n \to L$ .

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RemarkDependence on completeness

The proof crucially uses the completeness axiom (existence of suprema). Over $\mathbb{Q}$ , the MCT fails: the sequence $a_n = (1 + 1/n)^n$ is increasing and bounded above in $\mathbb{Q}$ , but does not converge to a rational (it converges to $e \notin \mathbb{Q}$ in $\mathbb{R}$ ).

Applications

ExampleConvergence of (1 + 1/n)^n

Define $a_n = (1 + 1/n)^n$ . One can show $(a_n)$ is increasing and bounded above (by $3$ , say). By the MCT, $(a_n)$ converges. The limit is defined to be $e \approx 2.71828\ldots$

ExampleSeries with positive terms

Let $\sum a_n$ be a series with $a_n \geq 0$ . The partial sums $s_n = \sum_{k=1}^n a_k$ form an increasing sequence. By the MCT, $\sum a_n$ converges if and only if the partial sums are bounded above. If unbounded, $s_n \to +\infty$ .

ExampleSquare root iteration

Define $a_1 = 1$ and $a_{n+1} = (a_n + 2/a_n)/2$ . One can show $(a_n)$ is decreasing and bounded below by $\sqrt{2}$ . By the MCT, $a_n \to L$ for some $L$ . Taking limits in the recursion, $L = (L + 2/L)/2$ , so $L^2 = 2$ , giving $L = \sqrt{2}$ (Newton's method for square roots).

ExampleNested intervals theorem

Let $I_n = [a_n, b_n]$ be closed intervals with $I_{n+1} \subseteq I_n$ for all $n$ . Then $(a_n)$ is increasing and bounded above by $b_1$ , and $(b_n)$ is decreasing and bounded below by $a_1$ . By the MCT, $a_n \to \alpha$ and $b_n \to \beta$ with $\alpha \leq \beta$ . Thus $\bigcap_{n=1}^\infty I_n = [\alpha, \beta]$ is nonempty. This is equivalent to completeness.

Summary

The Monotone Convergence Theorem is a pillar of real analysis:

Every bounded monotone sequence converges (to its supremum or infimum).
It is equivalent to the completeness axiom.
Applications include defining $e$ , convergence of series, and iterative methods.

See Cauchy Criterion for an alternative characterization of convergence, and Lebesgue Monotone Convergence for the measure-theoretic generalization.