I came out with this proof for the Well Ordering principle but couldn't find anything similar on the internet and was wondering if it's wrong
Formal Question : Prove that the non-empty set S of natural numbers contains a least element
Proof with Induction
If S contains only one element, this element is both the largest and the least and so S
contains a least.
Let's assume for every set that contains n elements there is a least and prove that a set with
n+1 elements also has a least.
Let S be a set with n+1 elements and let A be a set with n random elements chosen from S such
that there is j that belongs to S but not to A.
We can infer from the induction's assumption A has a least because A is a set with n elements, call it k.
Because both k and j are natural numbers either k smaller than j or j smaller than k
and so S has a least which is whether k or j
Sounds right?