Intersection of Inductive Set as Subset of Real Numbers is Inductive Set

Theorem
Let $\mathcal A$ be a set of inductive sets.

Then their intersection is an inductive set.

Proof
By definition of inductive set:


 * $\forall S \in \mathcal A: 1 \in S$

Thus by definition of set intersection:
 * $\displaystyle 1 \in \bigcap_{S \mathop \in \mathcal A} S$

Also by definition of inductive set:


 * $\forall S \in \mathcal A: x \in S \implies x + 1 \in S$

So:

Hence the result, by definition of inductive set.