Show that f has a minimun

Question

been trying to solve this for some time now.

f is continuous in $ [0,\infty), $ and $\lim_{x\to \infty}f(x) = L . $ prove that if there exist $x \ge 0 $ such that f(x) < L then f has a minimum in $ [0,\infty)$.

Answer 1

There exists some $y\in[0,\infty)$ such that $f(t)>f(x)$ for $t\ge y$. Then $f$ meets a minimum in $[0,y]$.