Showing that $E[X|XI would like to show that:
$\hspace{2mm} E[X|X<x] \hspace{2mm} \leq  \hspace{2mm} E[X] \hspace{2mm} $ for any $x$ 
X is a continuous R.V. and admits a pdf. I'm guessing this isn't too hard but I can't come up with a rigorous proof. Thanks so much. 

Hint: $$E[X] = E[X|X<x]P(X<x) + E[X|X\geq x]P(X\geq x)$$
Also:
$$E(X|X<x)< x \leq E(X|X\geq x)$$
