Prove the following theorem
If G is a collection of induction sets, then the intersection of all the sets in G is an induction set.
Since all $\displaystyle G_i$ are all inductive $\displaystyle 1\in G_{1}\wedge 1\in G_{2}\wedge 1\in G_{3}$ and............................................$\displaystyle 1\in G_{n}$ =====> $\displaystyle 1\in\bigcap G_i$.
Suppose now $\displaystyle k\in\bigcap G_i$ ====> $\displaystyle k\in G_{1}\wedge k\in G_{2}\wedge k\in G_{3}$ and...........................................$\displaystyle k\in G_{n}$====>$\displaystyle k+1\in G_{1}\wedge k+1\in G_{2}\wedge k+1\in G_{3}$ and............................................... ..$\displaystyle k+1\in G_{n}$=====>$\displaystyle k+1\in\bigcap G_i$
Hence $\displaystyle \bigcap G_i$ is inductive