Consider the following fragments from Takesaki's second volume "Theory of operator algebras" in chapter VII on weight theory, lemma 1.15. Here $\mathcal{M}$ is a von Neumann algebra and $\mathscr{S}$ is the unit ball of $\mathscr{M}$.

Why exactly can we approximate a point in the strong closure of $E^*p\cap r\mathscr{S}$ by a sequence (as opposed to a net)? I imagine it has something to do with metrizability of $p\mathscr{M}p$? Another explanation is that $\mathscr{M}_p$ is a typo and it should be $\mathscr{M}p \cap \mathscr{S}$ that should be metrizable? But in that case I don't see why this must be metrizable.

Any help is highly appreciated!