Cantor's intersection theorem: Difference between revisions

Revision as of 15:00, 9 July 2022

Revision as of 15:00, 9 July 2022

Line 49:

:<math>x_k \le x_{k+1}</math>,

so <math>(x_k)_{k \geq 0}</math> is an increasing sequence contained in the bounded set <math>C_0</math>. The [[monotone convergence theorem]] for bounded sequences of real numbers now guarantees the existence of a [[Limit of a sequence|limit point]]

:<math>x=\lim_{k\to \infty} x_k.</math>

For fixed <math>k</math>, <math>x_j\in C_k</math> for all <math>j\geq k</math>, and since <math>C_k</math> is closed and <math>x</math> is a [[limit point]], it follows that <math>x\in C_k</math>. Our choice of <math>k</math> is arbitrary, hence <math>x</math> belongs to <math>{\textstyle \bigcap_{k = 0}^\infty C_k}</math> and the proof is complete. ∎

== Variant in complete metric spaces ==