Let $p(n)$ denote the probability that a randomly chosen one-to-one function $f:\lbrace 1,2,3...,n \rbrace \rightarrow \lbrace 1,2,3...,n \rbrace$ has a least one fixed point. (least one integer $k$ for which $f(k)=k, 1\leq k \leq n$)
Prove that $$\lim_{n \to \infty} p(n) = 1-\frac{1}{e}$$