Prove that any uncountable subset of the Euclidean n-space contains an countable subset with the property that the distances between different pairs of points are different (that is, for any points P_1 \not\equal{} P_2 and Q_1\not\equal{} Q_2 of this subset, \overline{P_1P_2}\equal{}\overline{Q_1Q_2} implies either P_1\equal{}Q_1 and P_2\equal{}Q_2, or P_1\equal{}Q_2 and P_2\equal{}Q_1). Show that a similar statement is not valid if the Euclidean n-space is replaced with a (separable) Hilbert space. inductiongeometry3D geometryspherereal analysisreal analysis unsolved