Limit of continuous function with lower bound but without a minimum when $x \to \infty$

Question

Let $f: [0, \infty) \to R$ continuous function, that has lower limit:

$m = inf \{ f([0, \infty))\}$

but doesn't have global minimum.

Is it true that $lim_{x \to \infty}=m$? If yes, how to prove that?

Answer 1

You can prove by contradiction that if $$lim_{x \to \infty}\neq m$$

then by EVT $f$ should have a global minimum.