Mimic the first part of the proof of Theorem $1$ from this week$$s notes to complete the proof. The book reference is Theorem $2\mathrm{.}2$

For a subset S$,$ define the seat 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$\}.$

Show that if $1$ does not exist in X$,$ then S has the smallest element.