Problem 2

Show that

r=2

is the largest real number

r

that satisfies the following condition: if a sequence

a_1,a_2,\ldots

of positive integers fulfills the inequalities

a_n\le a_{n+2}\le\sqrt{a_n^2+r\,a_{n+1}}

for every positive integer

n

, then there exists a positive integer

M

such that

a_{n+2}=a_n

for every

n\ge M

Step 2 of 6: First lemma for r=2

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a_{n+1}\le a_n \implies a_{n+2}=a_n

Detailed analysis

Now take $r=2$ . If $a_{n+1}\le a_n$ , then $a_{n+2}\le\sqrt{a_n^2+2a_{n+1}}\le\sqrt{a_n^2+2a_n+1}=a_n+1$ . Since $a_{n+2}$ is an integer with $a_n\le a_{n+2}<a_n+1$ , we must have $a_{n+2}=a_n$ .