Suppose that the $\beta=\liminf\limits_{n\to\infty}{a_n}>0$ where $\{a_n\}\subset(0,\infty)$ then there exists $a\in \mathbb{R}$ such that $a_n\geq a>0$ for all $n\in \mathbb{N}$.
I've always known by the definition of $\liminf$ that, for $\epsilon=\beta/2>0,$ there exists $n_0$ such that $$a_n>\beta/2,\;\;\forall\;n\geq n_0.$$
However, I can't really figure out the existence of $a>0$ such that $a_n\geq a>0$ for all $n\in \mathbb{N}$. Can anyone please, explain this to me? Thanks for your time!
Note: This statement can be found in G. Zamani Eskandani and M. Raeisi, page 12.
With the correct hypotheses this question is straightforward. You've noticed that there is an $n_0$ and an $a'$ such that $n \geq n_0$ implies that $a_n > a' >0$. Now just take $a = \min\{a_1, \dots, a_{n_0-1}, a'\}>0$.