Checking Inductive problem

[cooldog182]

Hi, this proof by induction is different to what i'm normally used to. Could you see what i've done and see if it's right please?

QUESTION:
Prove by induction, that for all natural numbers n>1

'If n people stand in a line, and if the first person in the line is a woman and the last person in the line is a man, then somewhere in the line there is a woman directly infront of a man'

ANSWER:

Basis Step:
P(1) = Woman
P(n) = Man

Inductive Step:
Suppose P(1), P(2), P(3)................P(n) is true

We have to verify that P(n+1) is true

P(n+1) is certainly true when n+1 is a woman and n+2 is a man

If n+1 is not a woman, then it must be a man. Therefore for all n, the same is true, leading the whole argument to be true.

 

[pka]

Cool, I suspect that this particular problem is meant to be worked using the basic principle of induction, well ordering. That is the root source of induction: each nonempty subset of positive integers has a first member, a least element.
Suppose, J={j: a man is in the jth place in line}; J is a subset of the positive integers. We know that J is not empty because we are given that there is a man in the nth place so that means that n belongs to J. Using the ‘well ordering’ of the positive integers, J has a first or least term call it k. That means that k is in J or a man is in the kth position in line. We know that k≠1 because we are given that a woman is in the first position so 1 is not in J. Therefore, k-1 is a positive integer from 1 to n-1 and because k-1<k, k-1 is not in J (k is the least in J) a woman must be in that position Thus we have a woman in the (k-1)th position and an man in kth position.