Prove or disprove this statement. If G is a group in which every proper subgroup is cyclic, then G is cyclic

Question

I want someone to check my proof. Suppose every proper subgroup is cyclic,but G is not cyclic. Since subgroup is cyclic,It generator must exist in subgroup. Then all element in subgroup exist in G;it included the generator of subgroup. then G has generator and cyclic contradict my assumetion therefore if every proper subgroup is cyclic G is cyclic