Suppose that $f$ is continuous on $[a,b]$. Prove that given $\epsilon>0$, there exist points $x_0=a
$E_k=\{y: f(x)=y\ for\ some\ x \in [x_{k-1}, x_k]\}$,
then sup$E_k -$inf$E_k<\epsilon$ for $k=1,2,...,n$.
Just from looking at the question, I suspect that I need to eventually the Bolzano Weierstrass Theorem and Extreme Value Theorem to complete this proof. However, I am having some difficulty setting up the problem so that I can use theorems as needed. Please help! I really appreciate any guidance that I can get.