Complete Induction: Proving using WOP/Induction

[jgens]

Homework Statement

Prove the principle of complete induction using either the well-ordering principle or ordinary mathematical induction.

Homework Equations

N/A

The Attempt at a Solution

Suppose that A is a collection of natural numbers such that 1 \in A and n+1 \in A whenever 1, \dots, n \in A. Now suppose that B is the collection of those natural numbers not in A. If B \neq \emptyset, then by the well-ordering principle, B must have some least element. Clearly, this least element cannot be 1 so suppose instead that k+1 is the least element in B, in which case 1, \dots, k are all in A; however, this implies that k+1 \in A which is a contradiction. Thus our assumption that B was non-empty must have been incorrect and A = \mathbb{N}.

Is this correct?

 

[jgens]

Since it's been roughly a day, does anyone have feedback?