A rich knight has a chest and a lot of coins, so every day he puts into the chest some quantity of coins - among the numbers $1, 2, \ldots, 100$. If there exist two days on which he added equal quantities of coins (say, $k$ coins) and he has added in total at most $100k$ coins on the days between these two days, he stops putting coins into the chest. Prove that this will necessarily happen eventually.