Every graph is a good graph. The proof is given below.
Suppose p be an maximal path in the graph G.
Let u and v be its end points.
Since p is the maximal path in the graph G all the neighbours of u and v will be on the path p.
So, if we remove either u or v from the path p then the neighbours of u or the neighbours of v remain connected through the path p.
Hence, after removing u or v from G the graph G remains connected.
Hence, for any graph we will get at least two vertices which are not cut vertices.
Hence every graph is a good graph (proved).
Now, in the question it is asked to prove the proposition
PROPOSITION :- A subgraph H of G is a good graph iff G is a good graph.
Now, according to the theorem ( Every graph is a good graph) we have both sides of the above proposition as true. (That is both sides of iff).
So, the above proposition is always true (proved).