The cantor's intersection theorem states that if $X$ is a complete metric space, then, A decreasing nested sequence of non-empty compact subsets of has a non-empty intersection.
Consider the nested sets : $[-r,r]$ where $r \in (0,1]$. Then , clearly, $[-r,r]$ forms a decreasing diameter nested sequence of sets but whose intersection is empty as $r$ doesn't take the value $0$. Does this contradict the cantor's intersection theorem? Where could I be wrong? Thanks!