Hi

I am here after a long time. I want to prove that no natural number can be both even and odd. This is from basic analysis book. So I start by saying that let S be the set of natural numbers such that they are both even and odd.

I assume the negation. So let S be non-empty. Since S is a subset of

, by well ordering principle, S has a least element

. So

is both odd and even.

and

for some

. Which means

.

. Which means that

is an even number. At this point can I say that I have reached a contradiction. Or do I first need to prove tthathat

is an odd number to use that property