Two examples which come to my mind are:
$(\forall n \in \mathbb{N})$ $\, \,$ $p_{n+1}<2^{2^{n}}$.
For every $n \in \mathbb{N}_{\geq 12}$, $\,$ $p_{n}>3n$.
By the way, Erdös's proof of Bertrand's postulate is not by induction (it depends on some results which can be proven via mathematical induction, though).