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For questions concerning limit superior and limit inferior of sequences (of real numbers and various generalizations) and also for $\limsup$ and $\liminf$ of sequences of sets.

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Comparing two limsup's

No, this equality does not hold in general. What is true is that rhs is at most twice as large as lhs and this is sharp. For the sake of brevity I will only show an example where lhs is $1$ and rhs …
Martin Sleziak's user avatar