Theorem: If $f$ is continuous on $[a,b]$ and $f(a)<0<f(b)$, then there is some number $x$ in $[a,b]$ such that $f(x)=0.$
Problem 3
b) The proof of the Theorem 7.1 *(the one above)*depended upon considering $A=\{x:a\leq x\leq b $, and $f(x)<0$ on $[a,x]$$ \}$ Give another proof which depends upon consideration of $B=\{x:a\leq x\leq b , f(x)<0\}$. Which point x in $[a,b]$ will this proof locate? Give an example where the sets A and B are not the same.
What I did:
I did very little on this part of the problem, only could come up with an example where $A\neq B$, which basically was a sine function where the point $a$ is in the region $\sin(x)<0$ and $b$ is in a region where $\sin(b)>0$, and $b-a>2\pi$.