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 3 of 6: Second lemma for r=2

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

Detailed analysis

Similarly, if $a_n\le a_{n+1}$ , then $a_{n+2}\le\sqrt{a_n^2+2a_{n+1}}\le\sqrt{a_{n+1}^2+2a_{n+1}+1}=a_{n+1}+1$ ; the same computation as in the first lemma with the roles of $a_n,a_{n+1}$ swapped shows directly $a_{n+2}\le a_{n+1}$ .