TSTST 2024 Problem 7

An infinite sequence $a_{1},a_{2},a_{3},\ldots$ of real numbers satisfies

$$a_{2n - 1} + a_{2n} > a_{2n + 1} + a_{2n + 2} \mathrm{and} a_{2n} + a_{2n + 1}< a_{2n + 2} + a_{2n + 3}$$

for every positive integer $n$ . Prove that there exists a real number $C$ such that $a_{n}a_{n + 1}< C$ for every positive integer $n$ .