If $C_r^*(G)$ is purely infinite then every non-zero positive element of $C_0(G^{(0)})$ is infinite in $C_r^*(G)$

Question

Here $G$ is a groupoid and $G^{(0)}$ is its unit space. The statement seems trivially true but I don't see why exactly.

Edit: Definitions:

1. Infinite projection - A projection $p$ in a C*-algebra is infinite if it is Murray-vN equivalent to a proper subprojection. Equivalently, it is infinite if there exists a non-zero positive element a with $p\oplus a\precsim p$ where $\precsim$ means "Cuntz below"
2. Infinite positive element - Same as the last part of the above definition.
3. Purely infinite - A C*-algebra $A$ is purely infinite if, for each non-zero positive element $x$, the hereditary subalgebra $\overline{xAx}$ contains an infinite projection.

I don't know how to show every non-zero positive element in that subalgebra is a projection, let alone infinite. I don't how the two can be related. I know that every positive element is self-adjoint.