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 that is an odd number to use that property ?