The bookUnit 1 · The Real Line3 / 83

limsup, liminf and subsequential limits

Why this is asked: Compute limsup/liminf of explicit sequences fast, and know the algebra: limsup(aₙ + bₙ) ≤ limsup aₙ + limsup bₙ, with equality if one converges. Convergence ⇔ limsup = liminf (finite).

limsup and liminf — three equivalent definitions and the algebra

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Computing limsup and liminf without guessinginteractive

A worked sequence, the subsequential limit set, and the inequality that is only ever one-way.

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The trap here

“ $\limsup(a_{n} + b_{n}) = \limsup a_{n} + \limsup b_{n}$ ” — false

$a_{n} = (-1)^{n}, b_{n} = (-1)^{n}$ ⁺ $^{1}$

$a_{n} + b_{n} \equiv 0$ , so the left side is 0 while the right side is 1 + 1 = 2.