Hey I just found a solution on my own.  
Let $a$ be an arbitrary Element of a Hilbert space.  
We have  
$\Vert P_2 P_1 a\Vert^2=\langle P_2 P_1a,P_1a\rangle=\Vert P_2 P_1 a\Vert\Vert P_1a\Vert\langle \frac{P_2P_1a}{\Vert P_2 P_1 a\Vert},\frac{P_1a}{\Vert P_2 P_1 a\Vert}\rangle<\epsilon\Vert P_2 P_1 a\Vert\Vert P_1a\Vert$.  
$\implies \Vert P_2 P_1 a\Vert<\epsilon\Vert P_1a\Vert$   
$\implies\Vert P_2 P_1 a\Vert^2<\epsilon^2\Vert P_1a\Vert^2$  
$\implies\langle P_1P_2P_1a,a\rangle\le\epsilon^2\langle P_1a,a\rangle$.  
Since   
$-\epsilon\Vert P_1a\Vert^2<-\Vert P_2 P_1a\Vert^2<\Vert P_2 P_1a\Vert^2$,   
we also have  
$-\epsilon^2\langle P_1a,a\rangle\le\langle P_2P_1P_2a,a\rangle$.  
Changing the roles of $P_1$ and $P_2$ yields the claim.