Show that every induced subgraph of a bipartite graph is bipartite.
My try:
Let $G'(V',E')$ be a subgraph of $G(V,E)$.
We know by hypothesis that $V=V_x\cup V_y$ then $V'\subset V_x \cup V_y$
So there are three cases:
Case I: $V'\subset V_x$ and $V' \not \subset V_y$
Then $E'= \emptyset$ and $G'$ would be a bigraph.
Case II: $V'\not \subset V_x$ and $V' \subset V_y$
Analogous to the previous case.
Case III: $V'\subset V_x$ and $V' \subset V_y$
Here is where I'm stock. I know that subgraph would be bipartite but I don't know how to argue it.
The simplest strategy is a different one. It is well-known (and easily shown if you didn't know it) that a graph is bipartite if and only if it doesn't contain a cycle of odd length. Since the subgraph of a bipartite graph obviously couldn't gain a cycle of odd length, it must also be bipartite.