I want to prove the following statement:
In well-ordered set $\langle\mathbb{N},<\rangle$, moving $0,1,2,3,...,n-1$ to the end, retaining that order, results in a well-ordered set $\langle \Bbb N,<^{(n)}\rangle$.
My work:
Saying that the relation obtained is $<^{(n)}$. I proved that the relation is a total order, but how one can prove that to every non-empty subset of $\langle\mathbb{N},<^{(n)}\rangle$ has least element (first element).
Thank you.