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Is this true?

$lim sup (a_{n} + b_{n}) = lim sup a_{n} + lim sup b_{n}$

No — it is false.

The counterexample

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

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

The kind of mistake this is

Limit assumed to exist

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

Drill statements like this

sequences limsup

Others that fail the same way

From The Real Line › limsup, liminf and subsequential limits

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