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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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Understanding a proof about limit of a sequence of open sets
We are reading a proof about the following limit
\begin{equation}\tag{1}
\lim_{n \to \infty} \sigma_1(T_n)= \sigma_1(T),
\end{equation}
where $T:D(T) \subseteq H \to H$ and $T_n:D(T_n) \subseteq H \to …