Principle of Mathematical Induction for Natural Numbers in Real Numbers

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Theorem

Let $\left({\R, +, \times, \le}\right)$ 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 $\R$:

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

where $\mathcal I$ 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:

$A = \N$.

$\blacksquare$


Sources