# Proving a limit diverges if $f$ is not bounded above

The question is as follows:

Suppose $$D \subset \mathbb{R}$$ and $$f:D\rightarrow \mathbb{R}$$ is monotone increasing. Let $$a \in \mathbb{R}$$ be a limit point of $$D\cap \left ( -\infty,a \right )$$. If $$f$$ is not bounded above, prove that $$\lim_{x\rightarrow a^{-}}f(x)=\infty.$$

This seems rather obvious. I know that $$\lim_{x\rightarrow a^{-}}f(x)=L \in \mathbb{R},$$ if given any $$\epsilon>0$$, $$\exists \ \delta > 0$$ such that if $$0 and $$x \in D$$, then $$\left | f(x)-L \right |<\epsilon.$$ But since $$f$$ is not bounded above, $$\sup_{x \in D\cap \left ( -\infty,a \right )}f(x)$$ does not exist. How can I continue from here?

• If you take $D = (-\pi/2,\pi/2)$, $f(x) = \tan x$, and $a = 0$, then the claim is not true. Seems like the hypotheses need to be sharpened. – Alex Ortiz Nov 2 '18 at 1:57
• You perhaps mean that $a$ is the largest limit point of $D$. – Paramanand Singh Nov 2 '18 at 2:45