I am stuck in a proof in Davidson's "$C^*$ algebras by examples" book. In section II.3, he proves that any abelian Von Neumann algebra $N$ on a separable Hilbert $H$ has a non-zero projection with uniform multiplicity. In doing so, he starts as follows :
Choose a sequence of unit vectors $x_n$ and projections $Q_n$ (in $N'$, the commutant) onto $Nx_n$ such that $x_n$ is separating for $N_{ | (\sum_{i=1}^{n-1}Q_i)^\perp}$. Let $\overline{Q_n}$ be the central cover of $Q_n$. Then $\overline{Q_n}\geq \overline{Q_{n+1}}$, the stated reason being that $x_n$ was chosen separating for the compression algebras.
What I don't understand is why one has $\overline{Q_n}\geq \overline{Q_{n+1}}$? (for the given reason or another)
Thanks for any help.