EDB — 0VS

Exercises

1. [0VS] Let \((X,d)\) be a metric space and let \(D⊆ X\), show that these clauses are equivalent:

   • \(D\) is not totally bounded;

   • there exists \(\varepsilon {\gt}0\) and there is a sequence \((x_ n)_ n⊆ D\) for which

     \[ ∀ n,m∈ℕ, ~ d(x_ n,x_ m)≥ \varepsilon \quad . \]