Balti Tee 1997 · Ülesanne 2

Solution:

Let $\ell$ be the least index such that $a_{\ell}>a_{1}$. Since $2 a_{\ell}-a_{1}$ is a positive integer larger than $a_{1}$, it occurs in the given sequence beyond $a_{\ell}$. In other words, there exists an index $m>\ell$ such that $a_{m}=2 a_{\ell}-a_{1}$. This completes the proof.