The goal of the following parts is to prove the following. $$Every non$-$empty subset S of N has an smallest

element$.$ This is called the well$-$ordering principle.

For a non$-$empty subset S$,$ define the set X as follows: n is in X if and only if the elements $1,2,...,$ n are totally

contained in the complement of S$.$ That is$,$ S does not contain any of the element of the set $\{1,2,3,...,$ n$\}.$

$($a$)$ Show that if $1$ in X$,$ then S has a smallest element.

$($b$)$ Show that X $=$ N$.$

$($c$)$ Show that if n $$ in X$,$ then n $+1$ in X$.$

$($d$)$ Assume that $1$ in X$.$ Use $($b$)$ and $($c$)$ to conclude that S has a smallest element. Hint. If $1$ in X and

X $=$ N$,$ then the induction axiom allows you to extract some information