Two examples which come to my mind are:

1. $(\forall n \in \mathbb{N})$ $\, \,$ $p_{n+1}<2^{2^{n}}$.

2. 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, but that's a different thing): what Erdös actually does in his proof is compare lower bounds for the central binomial coefficients $\binom{2n}{n}$ with some upper ones which he obtains by means of Legendre's formula, the Erdös-Kalmár inequality, and the assumption that there are no primes in $(n,2n]$.