The manner in which mathematical induction is introduced in my textbook has me wondering about its proof:

The text then goes on to say that this principle will not be proven, and then provides an analogy of falling dominos.Theorem: The Principle of Mathematical Induction

Suppose that the following two conditions are satisfied with regard to a statement about natural numbers:

CONDITION I: The statement is true for the natural number 1.

CONDITION II: If the statement is true for some natural number k, it is also true for the next natural number k + 1.

Then the statement is true for all natural numbers.

What does the proof of this theorem look like?