Lesson 2

Theorem. The interval \([0,1]\) is in one-to-one correspondence with the set of infinite binary sequences.

Proof. Consider the unique infinite binary representation of a real number \(0 < x \le 1\): \(0{,}x_{1}x_{2}x_{3}\dotsc\). We associate with it the infinite binary sequence \((x_{1},x_{2},x_{3}, \dotsc)\). This will not be a bijection because there will be "unused" sequences with an infinite number of zeros at the end. But the set of such sequences is countable, so it does not change the cardinality (by the fifth point of the theorem above).◼