Lemma about the embedded segments. Mappings of sets

Definition

The sequence of embeded segments is called contractive, if the lengths of segments tend to zero.

Theorem

If the sequence of embeded segments is contractive, then there exists unique point, belonging to all segments.

Ptoof

Let it be two such points C₁ and C₂. Then b_n − a_n ≥ |C₁ − C₂| and the length of segments does not tend to zero.