Monotone Sequences

The central idea

A sequence is increasing if $a_{n+1}\ge a_n$ and decreasing if $a_{n+1}\le a_n$. It is monotone if one relation holds throughout or eventually. The Monotone Convergence Theorem states that an increasing sequence bounded above, or a decreasing sequence bounded below, converges.

Definitions, hypotheses, and notation

Eventually monotone behavior is enough for convergence because deleting a finite prefix does not affect the limit. Strict inequalities are unnecessary; repeated equal terms are compatible with monotonicity. For recursive sequences, a bound often ensures the recurrence stays in a domain where its update function preserves order.

The theorem proves existence of a limit without initially knowing its value. Once convergence is established, continuity of the recurrence function can justify passing to the limit. Multiple fixed points require initial conditions and bounds to identify the attainable one.

Conceptual meaning

Monotonicity prevents back-and-forth oscillation; boundedness prevents escape to infinity. Together they force terms toward the least upper bound or greatest lower bound. Either condition alone is insufficient.

A dependable method and decision rule

1. Compute $a_{n+1}-a_n$ or $a_{n+1}/a_n$ when terms are positive.
2. Establish one sign for the difference or compare the ratio with one.
3. Find an explicit upper or lower bound.
4. Invoke monotone convergence only after both properties are proved.
5. If a recurrence is given, solve the candidate fixed-point equation and justify which root is reached.

Fully worked example

Common mistakes and why they fail