Prove the number of decompositions is finite

Question

Let $a,n$ two positive integers. Prove:

1. There is a sequence $a_1, a_2, ... a_n$ of positive integers such that: $$1+ \frac 1 a = \prod_{k=1}^{n}\left(1 + \frac 1 {a_k}\right)$$
2. The number of the above sequences is finite.

(1) is easy to prove by taking $a_i = an + (i-1), 1 \le i \le n$, but I can't prove (2). I tried to use induction by $n$ but it doesn't seem to work.

Answer 1

2. Each $a_k$ has to be $>a$, otherwise $\prod_{k=1}^{n}\left(1+\frac{1}{a_k}\right)$ would be greater than $1+\frac{1}{a}$. On the other hand, by the AM-GM inequality $$\prod_{k=1}^{n}\left(1+\frac{1}{a_k}\right)\leq \left(1+\frac{1}{n}\sum_{k=1}^{n}\frac{1}{a_k}\right)^n $$ hence the average value of $\frac{1}{a_k}$ has to be $\geq -1+\left(1+\frac{1}{a}\right)^{1/n}\approx\frac{1}{na}$.
   In particular there are at most a finite number of sequences fulfilling 1.