Natural Numbers form Inductive Set/Proof 1

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:

$\N = \ds \bigcap \II$

where $\II$ is the collection of all inductive sets.

Suppose that $n \in \N$.

Then by definition of intersection:

$\forall I \in \II: n \in I$

Because all these $I$ are inductive:

$\forall I \in \II: n + 1 \in I$

Again by definition of intersection:

$n + 1 \in \N$

$\blacksquare$