Monotone Convergence Theorem

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A monotone bounded sequence climbs to its supremum

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Terms N

80

Radius ε

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Nondecreasing sequence aₙ = 1 − 1/n

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aₙ ↑ L and aₙ ≤ L ⇒ aₙ → L = sup{aₙ}

Monotonicity controls the direction of motion; a bound prevents the terms from drifting forever. Together, these two simple conditions turn the completeness of $\mathbb R$ into a convergence result, often used to establish limits without guessing them in advance.

Theorem

Let $(a_n)$ be a sequence of real numbers.

• If $(a_n)$ is nondecreasing and bounded above, then it converges to $\sup\{a_n:n\in\mathbb{N}\}$.
• If $(a_n)$ is nonincreasing and bounded below, then it converges to $\inf\{a_n:n\in\mathbb{N}\}$.

Equivalently, a monotone real sequence converges to a finite real limit if and only if it is bounded.

Proof for a nondecreasing sequence

Assume $(a_n)$ is nondecreasing and bounded above. Its range $A=\{a_n:n\in\mathbb{N}\}$ is nonempty and bounded above. By completeness of $\mathbb{R}$, it has a least upper bound $L=\sup A$.

Let $\varepsilon>0$. The number $L-\varepsilon$ is less than the least upper bound, so it cannot be an upper bound of $A$. Therefore, there is an index $N$ such that

$$L-\varepsilon<a_N\leq L.$$

For every $n\geq N$, monotonicity and the definition of $L$ give

$$L-\varepsilon<a_N\leq a_n\leq L.$$

Thus $|a_n-L|<\varepsilon$ for every $n\geq N$, which proves that $a_n\to L$.

The nonincreasing case follows by applying the same argument to $(-a_n)$, or by using the greatest lower bound of the range. Conversely, every convergent real sequence is bounded, so a monotone sequence has a finite limit exactly when it is bounded.

Example

For $a_n=1-\frac{1}{n}$ with $n\geq1$, the sequence is nondecreasing and bounded above by $1$. Its supremum is $1$, so the theorem gives $\lim_{n\to\infty}a_n=1$.

For comparison, $b_n=n$ is nondecreasing but unbounded and has no finite real limit. The upper-bound hypothesis is what rules out this kind of escape.

Takeaway

A monotone real sequence with the bound in its direction of motion converges to the corresponding extremal bound: the supremum when increasing, the infimum when decreasing.