# Principle of Mathematical Induction/Zero-Based

## Theorem

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

Suppose that:

$(1): \quad \map P 0$ is true
$(2): \quad \forall k \in \N: k \ge 0 : \map P k \implies \map P {k + 1}$

Then:

$\map P n$ is true for all $n \in \N$.

## Proof

Consider $\N$ defined as a Peano structure.

The result follows from Principle of Mathematical Induction for Peano Structure.

$\blacksquare$

## Also see

• Results about Proofs by Induction can be found here.