Waiting for a smaller draw

Draw $U_0 \sim \text{Uniform}(0,1)$. Then keep drawing independent $U_1, U_2, \dots \sim \text{Uniform}(0,1)$ until one of them is smaller than $U_0$. Let $N$ be the number of extra draws this takes. Find $P(N > n)$, $P(N = 1)$, and $E[N]$. Is $N$ finite?