Prove that there exists an irrational number between any two

Andres Durham

Andres Durham

Answered question

2022-06-01

Prove that there exists an irrational number between any two real numbers a and b.

Answer & Explanation

Asher Swanson

Asher Swanson

Beginner2022-06-02Added 2 answers

Let a , b R   , a < b. Then ( a , b ) is uncountable. The set of rational numbers, Q is countable, so any subset of it is also countable.
Now suppose ( a , b ) contained no irrational number. Then ( a , b ) would be an uncountable subset of Q which is a contradiction.
Felix Moore

Felix Moore

Beginner2022-06-03Added 1 answers

The interval ( a , b ) is uncountable since f ( x ) = ( x a ) ( b a ) is a bijection from ( a , b ) to ( 0 , 1 ).
There is an irrational number in ( a , b ) since the rationals themselves are countable.

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