Help to Understand the Proof of Extreme Value Theorem Below are sceen shots taken from Pugh's Book Real Mathematical Analysis. 
My question mainly is from the proof below. How does it follow that $M<M$ from $b=c$? Thanks for your help.


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*Notice that during the first part of Case 2, we prove that the least upper bound of $V_c < M$, which means that the least upper bound of $f$ on $[a,c]$ is strictly less than $M$. 

*But recall that at the beginning, we defined $M$ as the least upper bound of $f$ on the entire interval $[a,b]$. Using the same notation, this means $M$ is the least upper bound of $V_b$. 

*So, if $b = c$, then $V_b = V_c$, so the least upper bound of $V_b < M$ but also the least upper bound of $V_b = M$ by definition of $M$, so $M < M$.
