Natural Numbers form Inductive Set

Theorem
Let $\N$ denote the natural numbers as subset of the real numbers $\R$.

Then $\N$ is an inductive set.

Proof
By definition of the natural numbers:


 * $\N = \displaystyle \bigcap \mathcal I$

where $\mathcal I$ is the collection of all inductive sets.

Suppose that $n \in \N$.

Then by definition of intersection:


 * $\forall I \in \mathcal I: n \in I$

Since all these $I$ are inductive:


 * $\forall I \in \mathcal I: n + 1 \in I$

Again by definition of intersection:


 * $n + 1 \in \N$