VTRMC 2011 Problem 6

Let $S$ be a set with an asymmetric relation $<$ ; this means that if $a,b\in S$ and $a<b$ , then we do not have $b<a$ . Prove that there exists a set $T$ containing $S$ with an asymmetric relation $\prec$ with the property that if $a,b\in S$ , then $a<b$ if and only if $a\prec b$ , and if $x,y\in T$ with $x\prec y$ , then there exists $t\in T$ such that $x\prec t\prec y$ .