Injectiveness, surjectiveness, biuniqueness. Countability and uncountability of sets

Theorem

Set of real numbers on the segment [0; 1] is uncountable.

Proof

Suppose opposite: all numbers on the segment can be enumerated [0; 1] = {x₁, x₂, …, x_n, …}.

Patition segment . Choose that segment, which does not contain x₁. Patition it in three segments and choose that, which does not contain x₂ ets. Constructed system of embedded segments has common point x ∈ [0; 1] which does not equal to any x_n.