Let $(x\_i)\_i$ be a sequence of distinct numbers in $[0,1]$.
Note that $[0, 1] \setminus \{x\_1, \cdots, x\_{n-1} \}$ can be written as a disjoint union of non-empty and non-singleton intervals $C\_{n, k}$.
Let $M\_n \equiv \max\_k |C\_{n,k}|$.

How do I prove that $\limsup\limits\_{n \to \infty} n M\_n\geq 1/\ln 2$?