Conditional expectation with Cauchy-Schwarz Inequality

Question

Consider real-valued random variables $X$, $Y$, and $Z$; and a scalar, positive constant $k$. I want to prove the following \begin{equation} E[1|X+Y<Z<X+Y+k]E[X^2|X+Y<Z<X+Y+k]\ge E([X|X+Y<Z<X+Y+k])^2, \end{equation} where $E[\cdot|\cdot]$ is the conditional expectation operator. I have two questions. First, can we say the following: \begin{equation} E[1|X+Y<Z<X+Y+k]=\int_{x\in\mathcal{R}}\left[ \int_{y\in\mathcal{R}}\left[\int_{z=x+y}^{x+y+k}f_{Z|X,Y}(z|x,y)dz\right]f_{Y|X}(y|x)dy \right]f_X(x)dx ? \end{equation} Second, can we define an "event" as \begin{equation} A=X+Y<Z<X+Y+k ? \end{equation} By doing so, we need to prove that \begin{equation} E[1|A]E[X^2|A]\ge E([X|A])^2, \end{equation} which can be shown by invoking the Cauchy-Schwarz inequality.