OSN 2009 Problem 2

For any real $x$ , let $\lfloor x\rfloor$ be the largest integer that is not more than $x$ . Given a sequence of positive integers $a_1,a_2,a_3,\ldots$ such that $a_1>1$ and $$\left\lfloor\frac{a_1+1}{a_2}\right\rfloor=\left\lfloor\frac{a_2+1}{a_3}\right\rfloor=\left\lfloor\frac{a_3+1}{a_4}\right\rfloor=\cdots$$ Prove that $$\left\lfloor\frac{a_n+1}{a_{n+1}}\right\rfloor\leq1$$ holds for every positive integer $n$ .