I am reading "Introduction to Algorithms 3rd Edition" by CLRS.
The following exercise is Exercise 26.1-3 for section 26.1 "Flow networks".
In section 26.1, the authors give a graph-theoretic definition of flow networks, discuss their properties, and define the maximum-flow problem precisely.
And the next section (section 26.2 "The Ford-Fulkerson method") presents the Ford-Fulkerson method for solving the maximum-flow problem.
The section 26.1 contains no theorems at all.
26.1-3
Suppose that a flow network $G=(V,E)$ violates the assumption that the network contains a path $s\rightsquigarrow v\rightsquigarrow t$ for all vertices $v\in V.$ Let $u$ be a vertex for which there is no path $s\rightsquigarrow u\rightsquigarrow t.$ Show that there must exist a maximum flow $f$ in $G$ such that $f(u,v)=f(v,u)=0$ for all vertices $v\in V.$
There is no path $s\rightsquigarrow u\rightsquigarrow t$ for $u$ by assumption.
So, there is no path $s\rightsquigarrow u$ or there is no path $u\rightsquigarrow t.$
If there is no path $s\rightsquigarrow u$, then material from the source doesn't reach $u.$
If there is no path $u\rightsquigarrow t$, then material from $u$ doesn't reach $t.$
So, intuitively, I believe the following holds:
If there exists a maximum flow $f$ in $G$, there exists a maximum flow $f^{'}$ in $G$ such that $f^{'}(u,v)=f^{'}(v,u)=0$ for all vertices $v\in V$. (By the way, we don't know if there exists a maximum flow for any flow network $G=(V,E)$ because section 26.1 is a section for only definitions and doesn't contain any theorem.)
But I cannot prove this fact.
(1) Please tell me the proof of the above fact.
(2) Why do the authors write this exercise? What does the above fact tell us?