Prove the following theorem

Question

Prove the following theorem:

If $f$ is continuous and strictly monotone increasing, and $\lim_{x\to a^+}f(x)=-\infty$ and $\lim_{x\to b^-}f(x)=\infty$ then $f$ accepts each real value in $(a, b)$ exactly once

I assume I need to use the MVT.

Answer 1

Use the Intermediate Value Theorem. For any $c \in \mathbb{R}$, you can find $M,m \in (a,b)$, s.t. $f(m) < c < f(M)$. So as $f$ is continuous there exists $x_0 \in (m,M) \subset (a,b)$ s.t. $f(x_0) = c$