In order to prove that
a) $Y = E[X|\mathfrak{F}]$
iff
b) For all $Z \in L_2$: $\langle X-Y,Z \rangle = 0$
our script introduces $t \in \mathbb{R}$ and the proof of a)->b) ends with an argument that is based on taking the limit from both sides of $0$.
I understand everything up to the part where we get:
(*) $t \Vert Z \Vert_2^2 -2\langle X-Y,Z \rangle \geq 0$
But now we say that for $t \searrow 0 $ we get:
$\langle X-Y,Z \rangle \leq 0$
and for $t \nearrow 0$:
$\langle X-Y,Z \rangle \geq 0$
This last step confuses me. I can't figure out how we got from (*) to the last inequality. My mind is also a bit hazy on how this argument works in the first place. So if you could, please tell me how the last inequality comes to pass and maybe explain shortly how this argument works logically. Thank you
$(*)$ does not yield the proof. I think you already divided an inequlity by $t$ before arriving at $(*)$. That was a mistake. You cannot divide an inequality by a negative number (unless you change the inequality sign). So you have to use the fact that $(*)$ holds when $t >0$ and $t\|Z\|_2^{2}-2\langle (X-Y), Z \rangle \leq 0$ when $t <0$.