Principle of Mathematical Induction for Natural Numbers in Real Numbers

Theorem
Let $\struct {\R, +, \times, \le}$ be the field of real numbers.

Let $\N$ be the natural numbers in $\R$.

Suppose that $A \subseteq \N$ is an inductive set.

Then $A = \N$.

Proof
By definition of the natural numbers in $\R$:


 * $\N = \ds \bigcap \II$

where $\II$ is the set of inductive sets in $\R$.

Since $A$ was supposed to be inductive, it follows that:


 * $\N \subseteq A$

from Intersection is Subset: General Result.

Hence by definition of set equality:


 * $A = \N$