Is the following statement true? limsup(aₙ + bₙ) = limsup aₙ + limsup bₙ

No. aₙ = (−1)ⁿ, bₙ = (−1)ⁿ⁺¹ aₙ + bₙ ≡ 0, so the left side is 0 while the right side is 1 + 1 = 2.

Counterexample

Is this true?

$\limsup(a_{n} + b_{n}) = \limsup a_{n} + \limsup b_{n}$

No — it is false.

The counterexample

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

Why it fails

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

Keep practicing

Limit assumed to exist

Reasoning with a limit before establishing there is one. limsup is not lim.

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.