Theorem 2.43 states that any non-empty perfect set in the real vector space is uncountable.
Rudin uses basic induction to show that no point of $P$, where the points of $P$ are $x_1, x_2, \dots$ lies in the intersection of countably infinite sets. But as far as I know, basic induction can only prove a case for any natural number n and not infinity. And I'm pretty sure, in this case, Rudin is proving that for countably infinite points $x_n$, none of them lies in the intersection. A similar case is when induction alone can't prove that the countably infinite union of countably infinite sets is countable.