EDB — 0VR

Exercises

1. [0VR] Let be given a metric space \((X,d)\) and its subset \(C⊆ X\) that is totally bounded, as defined in [0V3]↺↻: show that \(C\) is bounded, i.e. for every \(x_ 0∈ C\) we have

   \[ \sup _{x∈ C} d(x_ 0,x){\lt}∞\quad , \]

   or equivalently, for every \(x_ 0∈ C\) there exists \(r{\gt}0\) such that \(C⊆ B(x_ 0,r)\).

   The opposite implication does not hold, as shown in [0VT]↺↻