updated: lim inf, lim sup and bertrand's postulate for a function

Question

Define $q:\mathbb{N}\rightarrow\mathbb{Q}^+$ with, $q(n)=min\{\alpha\in \mathbb{Q}^+\mid \exists \;p\in \mathbb{N},\; p \text{ prime and } p \in (n,\alpha n]\}$.

Is the $\liminf\limits_{n\rightarrow\infty} q(n)=1$?

I was thinking that for $p$ prime, $q(p-1)=1+\frac{1}{p-1}$ as $(1+\frac{1}{p-1})(p-1)=p$ is prime. Then this tends to $1$ for large primes $p$.

Also does Bertrand's postulate imply that $\liminf\limits_{n\rightarrow\infty} q(n)\leq 2$?

I'm just curious really, I was reading over bertrand's postulate and thought of this.

Answer 1

yes the limit is 1. can i ask you: why you need this ?