Can someone let me know whether the following theorem is written correctly? I based it off Apostal.
Let $S$ and $T$ be nonempty subsets of $\Bbb{R}$ such that $s\geq t$ for all $s$ in $S$ and $t$ in $T$. Then if $S$ has an infimum, then $T$ has an infimum and $\operatorname{inf}(S)\geq \operatorname{inf}(T)$.
It's false. Consider $S=[0,1]$ and $T=(-\infty,-20]$. Then, clearly holds that $s\geq t$ for all $s$ in $S$ and $t\in T$. Moreover, $\inf(S)=0$ but $T$ doesn't have an infimum.