EDB — 0BF

Exercises

1. [0BF]Let \(I_ n⊆ ℝ\) (for \(n∈ℕ\)) be closed and bounded non-empty intervals, such that \(I_{n+1}⊆ I_ n\): show that \(⋂_{n=0}^∞ I_ n\) is not empty.

   This result is known as Cantor’s intersection theorem [ 27 ] . It is valid in more general contexts, see [0VP]↺↻ and [0J6]↺↻.

   If we replace \(ℝ\) with \(ℚ\) and assume that \(I_ n⊆ ℚ\), is the result still valid?