For any function $g: \mathbb{R} \rightarrow \mathbb{R}$ and random variables $X, Y$, the "tower property" of conditional expectations is

$$E[g(X)]=E[E[g(X) \mid Y]] .$$

Provide a proof of this property when both $X, Y$ are discrete.

Let $U_{1}, U_{2}, \ldots$ be a sequence of independent uniform $U(0,1)$-random variables. For $x \in[0,1]$ find the expected number of $U_{i}$ 's needed such that their sum exceeds $x$, that is, find $E[N(x)]$ where

$$N(x)=\min \left\{n: \sum_{i=1}^{n} U_{i}>x\right\}$$

[Hint: Write $\left.E[N(x)]=E\left[E\left[N(x) \mid U_{1}\right]\right] .\right]$