EDB — 0NC

Exercises

1. [0NC] Let \(\varepsilon _ n{\gt}0\) be an infinitesimal decreasing sequence. If \((x_ n)⊂ X\) is a Cauchy sequence, there exists a subsequence \(n_ k\) such that

   \[ ∀ k ∈ℕ,~ ∀ h ∈ℕ,~ h{\gt}k ⇒ d(x_{n_ k},x_{n_ h})≤ \varepsilon _ k~ ~ . \]
   Solution 1

   [0ND]↺↻

   This property is often used by choosing \(\varepsilon _ n=2^{-n}\), or other sequence whose series converges.