Rational and Irrational numbers

Lemma 1

1. Between any two different rational numbers a and b there is at least one other rational number.
2. Between any two different irrational numbers a and b there is at least one other rational number.
3. Between any two different rational numbers a and b there is at least one other irrational number.
4. Between any two different irrational numbers a and b there is at least one other irrational number.

Corollary

In all four statements assertion of existence of a single number can be replaced with the assertion of existence of infinitely many numbers.

Lemma 2

Union of two countable sets is countable.

Lemma 3

Union of a countable number of countable sets is countable.

Lemma 4

The set Q of all rational numbers is equivalent to the set N of all integers.

Lemma 5

The set I of all irrational numbers is not countable.

Lemma 6

The set of all algebraic numbers is countable.

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