Principle of Mathematical Induction/Proof

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Let $\map P n$ be a propositional function depending on $n \in \Z$.

Let $n_0 \in \Z$ be given.

Suppose that:

$(1): \quad \map P {n_0}$ is true
$(2): \quad \forall k \in \Z: k \ge n_0 : \map P k \implies \map P {k + 1}$


$\map P n$ is true for all $n \in \Z$ such that $n \ge n_0$.


Let $\Z_{\ge n_0}$ denote the set:

$\Z_{\ge n_0} = \set {n \in \Z: n \ge n_0}$

Let $S$ be the set of integers defined as:

$S = \set {n \in \Z_{\ge n_0}: \map P n}$

That is, the set of all integers for which $n \ge n_0$ and for which $\map P n$ holds.

From Subset of Set with Propositional Function we have that:

$S \subseteq \Z_{\ge n_0}$

From $(1)$ we have that $\map P {n_0}$.

Hence $n_0 \in S$.

Let $k \in S$.

Then $\map P k$ holds.

But by $(2)$, $\map P {k + 1}$ also holds.

This implies $k + 1 \in S$.

So as:

$S \subseteq \Z_{\ge n_0}$


$S$ satisfies $(1)$ and $(2)$

it follows by the Principle of Finite Induction that $S = \Z_{\ge n_0}$.

Hence for all $n \ge n_0$, $\map P n$ holds.