Monotone Convergence Theorem — Mathematics Knowledge System

Overview

The Monotone Convergence Theorem (for sequences) states that every bounded monotone sequence of real numbers converges. If (aₙ) is increasing and bounded above, it converges to sup{aₙ}; if decreasing and bounded below, it converges to inf{aₙ}. This is one of the most direct consequences of the completeness of ℝ — the least-upper-bound property guarantees the supremum exists, and monotonicity forces the sequence to actually approach it. The theorem is a workhorse for proving convergence when the limit is not known in advance, such as showing (1+1/n)ⁿ converges (defining e) or establishing convergence of recursively defined sequences.

Intuition

An increasing sequence can only move in one direction — up. If it's also capped by some bound, it can't run off to infinity, so it has nowhere to go except settle down just below its 'ceiling.' That ceiling is exactly the supremum of the set of terms, and the theorem says the sequence must converge to it: since the sequence keeps climbing and can never overshoot the supremum, and since (by definition of supremum) it must eventually get arbitrarily close to it, convergence is forced. This is a much easier convergence criterion to apply than the general ε-N definition, because you never need to guess the limit — you just check monotonicity and boundedness.

Formal Definition

Definition

For a sequence (aₙ) of real numbers:

(a_n) \text{ increasing and bounded above} \implies a_n \to \sup\{a_n : n \in \mathbb{N}\}

Increasing case

(a_n) \text{ decreasing and bounded below} \implies a_n \to \inf\{a_n : n \in \mathbb{N}\}

Decreasing case

\text{monotone} + \text{bounded} \implies \text{convergent (relies on the least-upper-bound property of } \mathbb{R}\text{)}

Summary statement

Worked Examples

Show increasing: aₙ₊₁ - aₙ = (2 - 1/(n+1)) - (2 - 1/n) = 1/n - 1/(n+1) > 0.
$a_{n+1} - a_n = \frac{1}{n} - \frac{1}{n+1} > 0$
Bounded above by 2, since 1/n > 0 for all n means aₙ < 2 always.
$a_n = 2 - \frac{1}{n} < 2$
By the Monotone Convergence Theorem, aₙ converges to its supremum, which is 2 (since 1/n → 0).
$a_n \to \sup\{a_n\} = 2$

Answer: aₙ is increasing and bounded above by 2, so it converges, and the limit is 2.

Practice Problems

Difficulty 4/10

Show aₙ = (2n)/(n+1) is increasing and bounded above, and find its limit using MCT reasoning.

Difficulty 3/10

The Monotone Convergence Theorem for sequences fundamentally relies on which property of ℝ?

The Archimedean propertyThe least-upper-bound (completeness) propertyThe triangle inequalityThe commutativity of addition

Difficulty 6/10

Prove the Monotone Convergence Theorem for an increasing sequence bounded above.

Quiz

The Monotone Convergence Theorem states that a bounded monotone sequence of reals:

DivergesConverges to its supremum or infimumOscillatesMust be eventually constant

Why does the Monotone Convergence Theorem fail over ℚ?

ℚ has no bounded sequencesℚ lacks the least-upper-bound property, so a bounded set of rationals may have no rational supremumMonotone sequences don't exist in ℚIt doesn't fail — it holds over ℚ too

Summary

Every bounded, monotone sequence of real numbers converges — to its supremum if increasing, to its infimum if decreasing.
The theorem is a direct consequence of the completeness (least-upper-bound property) of ℝ, and fails over ℚ.
It is especially useful for proving convergence of recursively defined sequences without knowing the limit in advance.

References

WebsiteWikipedia — Monotone convergence theorem