If $||x|| < \sup\{||x_i||\}$ and $||x|| > \inf\{||x_i||\}$, then is $x \in \{x_i\}$?

Question

For any set $X \subset \mathbb{R}^d$, is it true that if $$||x|| < \sup\{||x_i|| : x_i \in X\} \quad\text{ and } \quad ||x|| > \inf\{||x_i|| : x_i \in X\}$$ then $$x \in X?$$

Answer 1

You can just see for $d=1$ this is not true. Say your subset is $\{1,3\}$. Then $x=2$ satisfies hypothesis but conclusion doesn't follow

Answer 2

Not necessarily. Let d=1, and X={1/n | n /in N} then 0=inf(X)<2/3