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, though).

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 $(n,2n]$.