In Arveson's book, the Kaplansky density theorem is proved in order to have this corollary: "Let $A$ be a self-adjoint algebra of operators on a separable Hilbert space $H$. Then for every operator $T$ in the strong closure of $A$, there is a sequence $T_n \in A$ such that $T_n \rightarrow T$ in the SOT" [The proof follows] "This corollary shows that in the separable case, the strong closure of a C*-Algebra of operators can be achieved by adjoining to the algebra all limits of its strongly convergent sequences."
And here I faint because I realize to have not understand at all what we did.
Isn't the strong closure of a C*-Algebra, by definition, the adjoining of all the limits of its strongly convergent sequences? Why a theorem is necessary?
The strong operator topology on $\mathscr{B}(H)$ is not metrisable if the underlying Hilbert space $H$ is infinite-dimensional. Therefore it need not be sequential a priori. What the Kaplansky density theorem really tells you is that elements in the SOT-closure of a C*-algebra are limits of (norm) bounded nets. Moreover, if the Hilbert space is separable, then bounded nets can be replaced with bounded sequences.