This problem might seem very elementary to someone. I am using the following arguments. I would like to make sure that it is right.
Let $f$ be a continuous function defined on $R$. If $\lim_{x\rightarrow\pm\infty}f(x)$ exists, then $f$ is a continuous function over the extended real line $R\cup\{\pm\infty\}$. Then we can say that there exists $x\in R\cup\{\pm\infty\}$ such that
$ f(y) \le f(x), \qquad \forall y\in R. $
The usual extreme value theorem is stated over a compact set $[a,b]$; see the wikipedia.
Thank you very much!
Anand