Prove that if x and y

ban1ka1u 2022-07-02 Answered
Prove that if x and y are irrational numbers, there exists an irrational number z such that y < z < x.
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Answers (1)

furniranizq
Answered 2022-07-03 Author has 20 answers
x y > 0 so ( x y ) 2 > 0. By the Archimedean property, there is a positive integer m such that 1 m < ( x y ) 2 . By increasing m if necessary, you can assume that m = 2 n 2 so that 1 2 n 2 < ( x y ) 2 . This means
1 2 n < x y 1 + 2 n y < 2 n x .
Now let k be the largest integer that is less than or equal to 2 n y. This implies
1 + k 1 + 2 n y < 2 n x .
But we also have 2 n y < k + 1, so
2 n y < k + 1 < 2 n x y < k + 1 2 n < x .
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