No, not quite, the conclusion is that given a group of k people where all people have the same age, then a group of k+1 people also must have the same age. Sinc

No, not quite, the conclusion is that given a group of k people where all people have the same age, then a group of k+1 people also must have the same age. Since we know that in a group of 1 all k people have the same age, then if the proof held then a group of 2 people would have the same as well, then 3, etc. This is a common tactic in induction, you make an assumption, prove that if true it implies a general result, and then give an explicit case where the assumption holds, proving the general result.