Prove that if f is continous on R and the limit as x approaches infinity and negative infinity is infinity, then f obtains its minimum value in R?

Question

Problem:

Suppose $f$ is a continous function defined on $\mathbb{R}$ s.t. $\lim\limits_{x\to -\infty}= \lim\limits_{x\to\infty} = \infty$. Then $f$ obtains its minimum value for some $x\in\mathbb{R}$.

My attempt:

1. Suppose $f$ is continous on $\mathbb{R}$ and $\lim\limits_{x\to -\infty}= \lim\limits_{x\to\infty} = \infty$.
2. Let $[a,b]\in\mathbb{R}$
3. Since $f$ is continous on $\mathbb{R}$ and $[a,b]\in\mathbb{R}$, then $f$ is continous on $[a,b]$.
4. Since $f$ is continous on $[a,b]$ and $[a,b]$ is closed and bounded, then $f$ is bounded on $[a,b]$.
5. Thus, $f$ obtains its maximum and minimum value at some $x\in [a,b]$

Here is where I am stuck because I know that the minimum may be to the left or right of this closed interval, but I do not know how to connect that idea and the idea of the limits that are given.

Thanks for your help.

Answer 1

Choose $N$ such that $f(x) >f(0)$ for $|x| >N$. On $[-N,N]$ $f$ attains its minimum value at some point $x$. Now verify that $f(x)$ is the absolute minimum of $f$ on $\mathbb R$.