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:

$A = \N$

$\blacksquare$

Sources

• 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers