We are given that $U_1, U_2,..., U_n$ are iid U(0,1), $S_n = \sum_{i=1}^n U_i$ and $N = min(k:S_k>1)$. We want to show that $P(S_k \leq t) = \frac{t^k}{k!}$.
I attempted to use induction, but I arrived at something rather wrong.
Here was my attempt.
Suppose it is true for k = 1: $P(S_1 \leq t) = \frac{t^1}{1!}$
Suppose it is true for k = n: $P(S_n \leq t) = \frac{t^n}{n!}$
Let's see what we then get for k = n + 1.
$P(S_n \leq t)P(S_n \leq t) = \frac{t^1}{1!}\frac{t^n}{n!} = \frac{t^{n+1}}{n!}$
But this is not what we want. Where is my error and how do I fix it?
