Monotone bounded sequences
Objective. Use monotone convergence and recursive bounds.
- 1. Learn
- 2. Worked example
- 3. Practice
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Learn: the key idea
A monotone bounded sequence is one that is either always increasing or always decreasing and stays within fixed bounds, and the Monotone Convergence Theorem guarantees such a sequence converges even without knowing the limit in advance. To apply it, first show the sequence is monotone (e.g. by comparing aₙ₊₁ − aₙ or using induction), then show it is bounded above (for increasing) or below (for decreasing). Once convergence to some L is known, recursive definitions aₙ₊₁ = f(aₙ) let you find L by solving L = f(L), since continuity of f means the limit passes through the recursion. For example, aₙ₊₁ = (aₙ + 2)/2 with a₁ = 0 is increasing and bounded above by 2, so it converges, and solving L = (L + 2)/2 gives L = 2.
Worked example
The recursive sequence aₙ₊₁ = (aₙ + 6)/2 with a₁ = 0 is monotone and bounded; find its limit.
- 1Show aₙ is increasing and bounded above by 6 using induction, so it converges to some L.
- 2Take limits of both sides of the recursion: L = (L + 6)/2.
- 3Multiply both sides by 2: 2L = L + 6.
- 4Solve for L: L = 6.
Answer: 6
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Start practisingCommon mistake. A common error is solving L = f(L) without first establishing that the sequence actually converges; a divergent or oscillating recursive sequence can satisfy the fixed-point equation while never approaching it.
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