I have a fairly basic inquiry but i would sleep better at night if i saw a proof of it.

Q: i know that if i take a connected subgraph with at least 2 vertices of any simple bipartite graph G that it has to be bipartite.

how would one go about proving that this is the case for any simple graph G.

i think that if G had vertices of all degree 2 i could prove it easily but above that i am not sure.

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Hint: If $G$ is bipartite, then its vertices can be split $V=V_1 \cup V_2$ so that the edges are subset of $V_1 \times V_2$.
Let $H$ be any subgraph of $G$ and let $V' \subset V$ be its vertices. Then you can split the vertices of $V'$
$$V'= (V'\cap V_1) \cup (V' \cap V_2)$$
and it is easy to see that this yields $H$ bi-partite.
P.S. if you are familiar with the fact that bipartite means two colorable , if you color the vertices of $G$ with two colors, that is a good coloring for any subgraph...