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The stability of an approximating sequence $(A_n)$ for an operator $A$usually requires, besides invertibility of $A$, the invertibility of furtheroperators, say $B, C, \dots$, that are well-associated to the sequence $(A_n)$.We study this set, $\{A,B,C,\dots\}$, of so-called stability indicators of$(A_n)$ and connect it to the asymptotics of $\|A_n\|$, $\|A_n^{-1}\|$ and$\kappa(A_n)=\|A_n\|\|A_n^{-1}\|$ as well as to spectral pollution by showingthat $\limsup {\rm Spec}_\varepsilon A_n= {\rm Spec}_\varepsilon A\cup{\rmSpec}_\varepsilon B\cup{\rm Spec}_\varepsilon C\cup\dots$. We further specify,for each of $\|A_n\|$, $\|A_n^{-1}\|$, $\kappa(A_n)$ and ${\rmSpec}_\varepsilon A_n$, under which conditions even convergence applies.

Finite sections: stability, spectral pollution and asymptotics of condition numbers and pseudospectra