Suppose that f is continuous on $(a,b)$ and $\lim_{x\to a^+} f(x) = \lim_{x\to b^-} f(x) = \infty$.
Prove that $f$ has a minimum on all of $(a,b)$.
I was thinking of applying the extreme value theorem, but I realized that it applies only to closed intervals. So I was thinking of defining a closed interval that is in $(a,b)$, and using the definition of one sided limits to go about the problem, but I don't know how to proceed. Help please?
Best Answer
Take a sequence of nested compact intervals contained in $(a,b)$ that 'converge' to $(a,b)$. What can you say about the minimum of $f$ in each of these compact intervals? What can you say about the values $f$ assumes outside these compact intervals?