The International Chess Federation is organizing an online chess tournament in which $20$ of the world’s top players will take part. Each player will play exactly one game against each other player. The tournament is spread over three weeks; it starts at $9$ a.m. on the Monday of Week $1$ and ends at $6$ p.m on the Friday of Week $3$. Note that before $9$ a.m. on the Monday of Week $1$ every player would have completed the same number of games in the tournament; namely, zero. Also, after $6$ p.m. on Friday in Week $3,$ every player would have completed the same number of games in the tournament; namely, nineteen.
Prove that at any point in time between $9$ a.m. on the Monday of Week $1$ and $6$ p.m. on the Friday of Week $3,$ there are at least two players who would have completed the same number of games in the tournament till that point.