Question

Supposons que pour chaque entier n∈N,A nsoit un sous-ensemble compact non-vide de R d, tel que A₁⊇A₂ ⊇⋯⊇Aₙ⊇…. . Montrez que l'intersection de tous ces ensemble est non videm c'est à dire ⋂[infinity]ₙ₌₁.Aₙ ≠

0 Indication : Le théorème de Heine-Borel n'est pas très utile ici. Il vaut mieux construire une suite bien choisit et extraire une sous-suite convergente en utilisant la compacité.

Answer 1

To prove that the intersection of these sets is non-empty, we can construct a well-chosen sequence and extract a convergent subsequence using the compactness property.

Step-by-step explanation:

We are given that for each positive integer n, An is a non-empty compact subset of Rd, such that A1⊇A2 ⊇⋯⊇An⊇…. We want to show that the intersection of all these sets is non-empty, that is, ⋂[infinity]ₙ₌₁.An ≠ 0. To prove this, we can construct a well-chosen sequence and extract a convergent subsequence using the compactness property.