Sufficient condition for a positive element to be properly infinite

Question

Consider a non-zero positive element $a$ of a simple C*-algebra $A$. Suppose it has a sub-projection which is Murray-vN equivalent to an infinite projection, i.e. there exists an infinite projection $h$ s.t. $h\precsim a$. Then is $a$ properly infinite?

Here are the steps and I don't see why exactly each one is true:

1. It is true that there always exists a non-zero positive $h$ which is Murray-vN equivalent to a subprojection of $a$.
2. If this $h$ is infinite, then the fact that $A$ is simple forces it to be properly infinite.
3. This forces $a$ to be properly infinite.

Help with any of the above points is appreciated.