If x3 is in B with x2, then we break the rule again. If there is an edge between to vertices in B, then our graph is not bipartite. In fact, if two vertices are next to each other in the cycle, then they can't be in the same partition. If x7 is in A, then we know that x6 and x8 are both in B. Vertices next to each other in the cycle are in different partitions. Now, we chose any cycle we liked, and so choosing a different one will not change things.
Every cycle is an even cycle.
We also chose any bipartite graph we liked, so the fact that there are no odd cycles has nothing to do with which bipartite graph we chose. What we have discovered holds for ALL bipartite graphs. So now we know that no bipartite graphs have any odd cycles. And that was what we were asked to show. We can use the proof to find a way to make the baseball teams we discussed before.