Alphabeta Math

limsup, liminf, and Subsequential Limits: Examples and Counterexamples

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✓ 9 results · all verified · 7 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 2 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Example

(-1)ᵏ has liminf = -1 and limsup = 1, so it does not converge

Example

The block sequence 1/1; 1/2, 2/2; 1/3, 2/3, 3/3; … has subsequential limit set exactly [0,1]

Example

aₖ = 2^-k + (-1)ᵏ has liminf aₖ₊₁/aₖ = 1/8, limsup aₖ₊₁/aₖ = 2 and lim aₖ^1/k = 1/2

Example

A positive sequence making all three inequalities of the ratio-to-root chain strict

Example

The four standard limits n^1/n → 1, a^1/n → 1, n^α/(1+p)ⁿ → 0 and xᵏ/k! → 0, computed

Counterexample

xₖ = (-1)ᵏ, yₖ = (-1)ᵏ⁺¹ give limsup(xₖ + yₖ) = 0 < 2 = limsup xₖ + limsup yₖ

Counterexample

xₖ = 1 + (-1)ᵏ, yₖ = 1 + (-1)ᵏ⁺¹ give limsup(xₖ yₖ) = 0 < 4

Counterexample

A sequence with limsup = +∞: the greatest subsequential limit exists only in ℝ̄

Counterexample

Null times divergent has no rule: xₖ = 1/k with yₖ = ck gives product limit c, and with yₖ = k² gives divergence

Sources