Each morning a student takes one of the three books he owns from his shelf. The probability that he chooses book $i$ is $a_i$, where $0 < a_i < 1$ for $i=1,2,3$ and the choices on successive days are independent.
In the evening he replaces the book at the left-hand end of the shelf. If $P_n$ denotes the probability that on day $n$ the student finds the books in the order $1,2,3$, from left to right, show that, irrespective of the initial arrangement of the books, $P_n$ converges as $n\to\infty$ and find the limit.