$\liminf_n a_n = \inf_n a_n$ if $a_n \ge a_m$ when $n\mid m$

Question

I would like to ask a reference for the following very easy result: can someone help?

Let $(a_n)_{n\ge 1}$ be a sequence of positive reals such that $a_m \le a_n$ whenever $n$ divides $m$. Then $$ \liminf_n a_n = \inf_n a_n. $$