Problem 5

Let

a_1,a_2,\ldots

be an infinite sequence of positive integers. Suppose there is an integer

N>1

such that, for every

n\ge N

, the number

\frac{a_1}{a_2}+\frac{a_2}{a_3}+\cdots+\frac{a_{n-1}}{a_n}+\frac{a_n}{a_1}

is an integer. Prove that there is a positive integer

M

such that

a_m=a_{m+1}

for all

m\ge M

Step 8 of 8: A descending chain of positive integers stabilizes

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a_{n+1}\mid a_n\text{ and }a_n\mid a_{n_0}\Longrightarrow a_n\text{ is eventually constant}

Detailed analysis

Choose $n_0$ after the common stabilization point. Then $a_n\mid a_{n_0}$ for every $n\ge n_0$ . Only finitely many positive divisors of $a_{n_0}$ exist, while $a_{n+1}\mid a_n$ ; the chain can therefore change only finitely often. Thus $a_n=a_{n+1}$ for all sufficiently large $n$ , as required.