We previously proved that $2$ lies in every interval $[a^{3},b^{3}]$ where MATH. These intervals $[a^{3},b^{3}]$ lie in MATH. However, a typical interval in MATH looks like $IJK$ where $I,J,K$ are intervals in $\QTR{cal}{C}2$. Show that $2$ lies in all of these as well. (Hint: If these intervals are all $>0$ then $IJK=[rup,svq]$ ($I=[r,s]$ etc). Show that $2\in IJK$ by using $a=\max (r,u,p)$ and $b=\min (s,v,q)$. When $I,J,K$ are not all $>0$, use the following:

Lemma. If $I=[r,s]$ is in $\QTR{cal}{C}2$ and $r\leq 0$, then MATH and $I^{\prime }$ is in $\QTR{cal}{C}2$.)