Answer

Note that \(E[X|Y]\) is a function of \(y\).

\[ \begin{aligned} E[E[X|Y]] &= \sum_{y} E[X|Y] p_Y(y)\newline &= \sum_{y} \sum_{x} xp_{X|Y}p_{Y}\newline &= \sum_{y}\sum_{x} xp_{X,Y}(x,y)\newline &= \sum_{x}x\sum_{y}p_{X,Y}(x,y)\newline &= \sum_{x} x p_{X}(x)\newline &= E[X] \end{aligned} \]