Existence minimal projections compact operators

Question

Consider the following fragment from Davidson's book on $C^*$-algebras:

Can someone explain why the sentence "Clearly, $P$ dominates a projection $E$ in $\mathfrak{A}$ with minimal positive rank." holds?

Answer 1

If that were not the case, you would have an infinite strictly decreasing sequence of subprojections of $P$, and then $P$ would have infinite rank.

Or, if the above is not obvious enough: start with $P$ of rank $n$. Is it minimal? If it is, take $E=P$. If it isn't, let $P_1$ be a proper subprojection of $P$. Then the rank of $P_1$ is at most $n-1$. Is it minimal? If it is, take $E=P_1$. If it isn't, take a proper subprojection $P_2$ of $P_1$. Then the rank of $P_2$ is at most $n-2$. If you repeat this for at most $n$ steps, either you found a minimal subprojection along the way, or you got to $P_n$ of rank 1. Either case, you have a minimal projection in $\mathfrak A$.