You just have to keep winning?

In a chess tournament, the organizer draws lots in order to pair off players at random with the loser in the previous match eliminated. This procedure repeats until one person remains.

In other words,

• if there are only 2 players in the tournament, then there must only be $2-1$ matches played,
• if there are only 3 players in the tournament, then there must only be $3-1$ matches played, and so on.

Is it true that for any positive integer $N,$ if there are only $N$ players in the tournament, then there must only be $N-1$ matches played?