4.12 Let I be the set of real numbers that are not rational; elements of II are called irrational numbers. Prove if a < b, then there exists x E I such that a < x < b. Hint: First show {r + √2 : r € Q} ≤ I.

Q: = Let a, b, and d be positive integers. Prove that if (a, b) : divides a, then (d, b) = 1.

A: Definition of gcd (a, b) =1 is a and b does not have common prime divisor. Using this we can easily…

Q: I = {x ∈ R : −4 ≤ x < 5}. Write I using interval notation. Hint: it might help to sketch this set on…

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Q: Let x and y be real numbers. Disprove: If |x| < |y| then |x+1|< |y+1]. Let x, y, and z be real…

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Q: Let a, be such that asb. Prove that tx Eqt ax cbx

A: Concept: We have a,b∈Q means that a, and b belongs to the rational numbers and a&lt;b means that a…