Monotone Convergence Theorem

A bounded monotone sequence converges to its supremum or infimum.

The idea

Theorem (Monotone Convergence Theorem).

A sequence of real numbers that is monotone — always increasing, or always decreasing — and bounded converges. An increasing sequence converges to its supremum, a decreasing one to its infimum.

The theorem supplies a limit without supplying a formula. Ordinarily we find a limit by simplifying $a_{n}$ until its value is visible; here two qualitative checks — monotone, bounded — guarantee that the limit exists, even when the terms are defined only by a recursion and no closed form is available.

Why it holds. Take an increasing sequence bounded above, by $M$ say. The set of terms is nonempty and bounded above, so completeness of $\mathbb{R}$ gives it a least upper bound $L = \sup a_{n}$. Let $\varepsilon > 0$. Since $L - \varepsilon$ is less than the least upper bound, it is not an upper bound at all, so some term $a_{N}$ exceeds it. The sequence increases and never passes $L$, so every term after $a_{N}$ lies between $L - \varepsilon$ and $L$. That is convergence to $L$.

Neither hypothesis can be dropped. An increasing unbounded sequence climbs past every bound, and a bounded sequence that is not monotone can oscillate forever.

Ways to work on it

• Walkthrough. State the theorem, find a limit, and see why boundedness is required.
• Practice. Find the limit of a bounded monotone sequence.
• Hardest. Find the limit of a recursively defined sequence.

Not sure where to start? Take the ten-question placement test.