Principle of Mathematical Induction/Zero-Based

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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.


Sources