An exercise about the positive operator

Question

Here is an exercise in functional analysis:

An operator $T$ on Hilbert space is positive is positive if and only if all compressions by finite-rank projections ($P_{n}TP_{n}$ for any $n$) are positive matrices.

I think the "only if" is easy to get because the positive operator $T=S^{*}S$. But how to prove the "if" of the exercise?

Answer 1

Take an $x\in H\setminus \{0\}$. You need to show that $\langle x,Tx\rangle > 0$. Can you see that a certain projection with one-dimensional range could be helpful?