Let $\left \{ X_r \right \}^\infty _{r=1} $ be a sequence of independent and identically distributed uniform on $[0,1]$ random variables. Let $0<x<1$ and define $$N(x) = \min \left \{ n \ge 1: X_1 + \cdots + X_n>x\right \}$$ (a) Show that: $$\mathbb{P}(N(x)>n)=\frac{x^n}{n!}$$
Recall that the moment generating function of a random variable $Y$ on the non-negative integers $\left\{0,1,2,...\right\}$ is defined by:$$\mathbb{E}[t^Y] = \sum ^\infty _{n=0} t^n \mathbb{P}(Y=n)$$ (b) Show that: $$\mathbb{E}[t^{N(x)}]=(t-1)e^{tx}+1$$
I am quite lost as to where to start. The only intuition I have would be that for part (a), that $$P(X_1>x)=1-x \\ P(X_1+X_2>x)=1-\frac{x}{2}\\ \vdots\\P(X_1+\cdots+X_n>x)=1-\frac{x}{n}$$and we can notice that $$\frac{x^n}{n!}=x \cdot \frac{x}{2}\cdot \cdots \cdot\frac{x}n$$ so I think that might be related but other than that I'm stuck
