$\lim \sup$ comparison question

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I want to figure out question 1.6 in Wheenden and Zygmund (2015)

The question is compare $\limsup_{k\to\infty} a_k$ with $\limsup(-\infty, a_k)$

(no more condition...)

I think $\limsup a_k \leq \limsup(-\infty, a_k)$

If $\limsup a_k$ is $+\infty$ or $a<\infty$ then equality holds.

If $\limsup a_k$ is $-\infty$ then inequality holds.

But I am not sure whether I'm right or not...