Prove the Intermediate-Value Theorem

Problem: 

Prove the Intermediate-Value Theorem. We assume that $f(x)$ is continuous on $[a,b]$ and that $f(a)\neq f(b)$. Prove that for every number $\Omega$ between $f(a)$ and $f(b)$ that there is at least one number $\omega$ in $(a,b)$ such that $f(\omega)=\Omega$.