Let $f:\mathbb R\to \mathbb R$ be continuous, $f(x)\to+\infty$ as $x\to\pm\infty$. Show that $f$ has a minimum.
If $f$ continuous and $\lim_{x\to-\infty }f(x)=\lim_{x\to\infty }f(x)=+\infty $ then $f$ takes its minimum.