If 9 people are seated in a row of 12 chairs, then at least one consecutive set of 3 chairs is occupied by people.
The most sparse set containing three empty chairs ("E") would appear to be SS E SS E SS E SS (11 chairs, 8 seated). If any of the three E were adjacent, the set would be shorter; if any of the seated ("S") were longer, the condition would have been satisfied. This seems obvious, but I do not know just what would count as a "proof". One could, of course, show all possible configurations or 9 S and 3 E -- that of course would prove the assumption.