Suppose G is a group, with subgroups H and K.
Prove that if H ∪ K is a subgroup of G implies that H ⊆ K or K ⊆ H.
I'm not really sure how to start this, I can prove that H ∩ K is a subgroup but I don't know how to approach the union. Any help is appreciated.
The question has been changed since I wrote this -- initially there was no requirement for $H$ or $K$ to be subgroups of $G$.
Unless I've misunderstood your question, what you have written is wrong. For example, take $G = \{0,1\}$ with the operation of addition modulo 2, $H = \{0\}$ and $K = \{1\}$. Then $H \cup K = G$ which is certainly a subgroup of $G$.