Question 98
a) Given any positive integer $n$, prove that every $n$ points in the closed unit square $[0, 1] \times [0, 1]$ can be joined by a path of length less than $2\sqrt{n} + 4$.
b) Prove that there exist $n$ points in the closed unit square $[0, 1] \times [0, 1]$ that cannot be joined by a path of length less than $\sqrt{n} - 1$.
b) Prove that there exist $n$ points in the closed unit square $[0, 1] \times [0, 1]$ that cannot be joined by a path of length less than $\sqrt{n} - 1$.