Putnam 1960 B5

Define a sequence $(a_n)$ by $a_0 =0$ and $a_n = 1 +\sin(a_{n-1}-1)$ for $n\geq 1$ . Evaluate $$\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^{n} a_k.$$