Monotone Sequences: Increasing, Decreasing, Bounded

A sequence is monotone if it never changes direction: increasing when each term is at least as big as the previous one, decreasing when each term is at most as big. The Monotone Convergence Theorem says a monotone sequence that is also bounded must converge — even when you cannot guess the limit in advance.

This matters because it proves convergence for sequences whose limits are famous constants: e = lim (1 + 1/n)ⁿ is defined exactly this way, and many recursive approximations rely on monotonicity plus boundedness.

Key Concepts

• Increasing: aₙ₊₁ ≥ aₙ for every n; strictly increasing if >.
• Decreasing: aₙ₊₁ ≤ aₙ for every n.
• Bounded above/below: all terms ≤ some M, or ≥ some m.
• Monotone Convergence Theorem: monotone + bounded ⇒ convergent.
• Limit = sup/inf: an increasing bounded sequence converges to the supremum of its terms.

Formulas

• Increasing: aₙ₊₁ ≥ aₙ for all n.
• Decreasing: aₙ₊₁ ≤ aₙ for all n.
• MCT: if (aₙ) is increasing and bounded above, then aₙ → sup{aₙ}.
• Fixed-point equation: for aₙ₊₁ = f(aₙ) with aₙ → L and f continuous, L = f(L).

Worked Example

Define a₁ = 1 and aₙ₊₁ = √(2 + aₙ). Compute: a₂ = √3 ≈ 1.7321, a₃ = √(2 + 1.7321) = √3.7321 ≈ 1.9319, a₄ = √3.9319 ≈ 1.9829 — the terms are increasing. Claim every term is below 2: a₁ = 1 < 2, and if aₙ < 2 then aₙ₊₁ = √(2 + aₙ) < √4 = 2, so by induction the sequence is bounded above by 2. By the Monotone Convergence Theorem it converges to some L ≤ 2. Passing to the limit in the recursion: L = √(2 + L), so L² = 2 + L, giving L² − L − 2 = 0, i.e. (L − 2)(L + 1) = 0. Since L ≥ 0, L = 2. The computed terms 1.7321, 1.9319, 1.9829 visibly climb toward 2, confirming the proof.

Common Misconceptions

• Monotone alone implies convergence. You need boundedness too — aₙ = n is increasing but diverges to infinity.
• The limit is always attained. The terms approach the supremum but may never equal it, as with 1 − 1/n → 1.
• Every convergent sequence is monotone. Not at all — (1 + (−1)ⁿ/n) converges to 1 while zigzagging.

Real-World Applications

• Defining e and π through monotone approximations.
• Recursive algorithms (square-root extraction, fixed-point iteration) converge by monotonicity.
• Economics: monotone bounded utility sequences model converging expectations.

Monotonicity plus boundedness guarantees convergence — the most practical convergence test in analysis. Whenever a recursion moves steadily in one direction, the Monotone Convergence Theorem finishes the job.