Limsup and Liminf

Key idea

Upper and lower limits of sequences.

In depth

lim sup aₙ is the largest limit of any subsequence (the eventual upper envelope); lim inf is the smallest. A sequence converges iff the two are equal and finite. They always exist for bounded sequences.

Worked example

For aₙ = (−1)ⁿ(1 + 1/n), lim sup = 1 and lim inf = −1.

Real-world uses

• Describing the highest and lowest long-run values of an oscillating system or a stock index

Common mistake

Thinking they are the sup and inf of the whole sequence.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Analysis & Topology

• Axioms of the Real Numbers – ℝ is a complete ordered field.
• Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
• Archimedean Property – For any real x there is an integer n > x.
• Density of Rationals – Between any two reals there is a rational.
• Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
• Limit Theorems for Sequences – Algebra of limits and squeeze theorem.