To prove the statement above we solve the problems below.
Starting at any vertex $A$ in any component of $G$, assign the color red to $A$ and proceed to color vertices along simple paths from $A$, alternating between red and blue. In that way every vertex in the component is eventually reached and colored.
In the procedure described above, the color assigned to a vertex $V$ depends on the length of the path followed from $A$ to $V$. A vertex reached by a path of odd length is colored blue, while a vertex reached by a path of even length is colored red.
(a) Let $A, V$ be vertices in a graph that contains no odd cycles and suppose that there are two paths from $A$ to $V$, one of odd length and one of even length. Show that this situation is impossible by deriving a contradiction.
(b) Prove if each component of a graph is bipartite, then the entire graph is bipartite. Then show that with the vertices of some component colored as described above, no edge has the same color assigned to both of its endpoints.
My question is what are we proving by solving the problem (a)? Are we proving that no two edges share the same endpoints?