Show that for any subgraph H of G, H is good if and only if G is good.

As written this claim is false. (Unless you define subgraph differently than what I am assuming)

Counter: Let G be (a path on 5 vertices). Then clearly G is good. Let H be a subgraph created by removing a single vertex of degree 2 in G. Then H is disconnected and cannot be good. Hence the claim fails.

This is true if you say: for any connected subgraph H of G, H is good if and only if G is good. But you already proved a stronger statement than this, that all connected graphs are good.