Given $f: \mathbf{R} \to \mathbf{R}$ is continuous and $\lim_{x\to -\infty}f(x) = \infty = \lim_{x\to \infty} f(x)$, show $f$ attains its minimum.

Question

There are plenty of questions on stack exchange similar to this but my question comes in a minute detail. My attempt: Let $a\in \mathbf{R}$ so that there are $c_1$, $c_2 \in \mathbf{R}$ such that $(c_1, \infty)\cap \mathbf{R} \neq \emptyset$ and $(-\infty, c_2)\cap \mathbf{R} \neq \emptyset$ and $f(t) \in (f(a), \infty)$ whenever $t\in (-\infty, c_2)\cup (c_1, \infty)$. Obviously, there's a standard extreme value argument if $c_2 <c_1$ but if $c_2\geq c_1$ then I come to the statement $$f(t) \in (f(a), \infty)$$ for each $t\in \mathbf{R}$. But this can not be true since $f(a) \notin (f(a), \infty)$ while $a\in \mathbf{R}$. Am I miss applying a definition somewhere?

Answer 1

The intuition is clear: Outside a compact interval $[-M,M]$ the function $f$ can be as large as you want. So the minimum of $f$ has to be inside the compact interval $[-M,M]$. Now use the fact that a continuous function achieves its minimum over a compact interval.