Here is a quotation of a book "C*-algebras and Finite-Dimensional Approximations" by Nate and Taka. (P245)
Let $H$ be a separable Hilbert space and $\Omega\subset B(H)$ be a separable set and let $F\subset \Omega$ be a finite set, $\chi\subset H$ be a finite set. If $Q$ is the orthogonal projection onto the span of $\chi$, then, by norm-compactness of the unit ball of $QH$, we can find a larger finite set $\tilde{\chi}\subset QH$ with the property that $||\tilde{P}Q-Q||<3\delta$, whenever $\tilde{P}$ is a finite-rank projection satisfying $||\tilde{P}(v)-v||<\delta$ for all $v\in \bar{\chi}$.
My question is why the author can find such a larger finite set $\tilde{\chi}$ that satisfy the condition ?? I mean the existence of $\tilde{\chi}$.
Since the unit ball of $QH$ is compact, we can find a finite set $\bar\chi\subset QH$ such that for every $w\in QH$ with $\|w\|\leq1$, we have some $v\in\bar\chi$ with $\|w-v\|<\delta$.
Then (recall that we assume that $\|\bar Pv-v\|<\delta$ for all $v\in \bar\chi$) for any $w\in H$ with $\|w\|\leq1$ we have $v\in\bar\chi$ with $\|Qw-v\|\leq\delta$, and $$ \|\bar PQw-Qw\|\leq \|\bar PQw-\bar P v\|+\|\bar P v - v\|+\|v-Qw\|\leq2\|Qw-v\|+\|\bar Pv-v\|<3\delta. $$ So $\|\bar PQ-Q\|<3\delta$.