# 7 please 6. Prove that if x is rational and y is irrational, then 2...
5. Prove that v6 is not rational (it is irrational)
5. Prove that v6 is not rational (it is irrational)
Question 5. Let x,y E R. Prove that if x and y are irrational, then at least one of 2 + y and c-y is irrational
Please help answer all parts!
(1) Prove that 75 is irrational. (State the Lemma that you will need in the proof. You do not need to prove the lemma.) (2) Disprove: The product of any rational number and any irrational number is irrational. (3) Fix the following statement so that it is true and prove it: The product of any rational number and any irrational number is irrational. (4) Prove that there is not a smallest real number greater than...
(3) (a) Prove that, between any two rational numbers, there is an irrational number (b) Prove that, between any two irrational numbers, there is a rational number
4. [5 Pts] Prove that the product of a non-zero rational number and an irrational number is irrational. Can you use a direct proof? Why or why not?
(10 points.) Recall that a real number a is said to be rational if a = " for some m,n e Z and n +0. (a) Use this definition to prove that if and y are both rational numbers, then r+y is also rational (b) Prove that if r is rational and y is irrational, then x+y is irrational
Let x,y ∈ R. Which of the following statements are true. If the
statement is true prove it, if not give a counterexample
a) If x is rational and y is irrational, then x y is irrational. b) If x and y are both irrational then x + y is irrational. c) Ifx and y are both irrational then ry is irrational d) Ifx is rational and y is irrational then ry is irrational.
(6) Prove that for all xeR, x > 0 there exists n eN such that 름 <z. (7) Prove that is irrational. (One or both numbers will be different, of course.)
(6) Prove that for all xeR, x > 0 there exists n eN such that 름
4. Define f(z) ={z. (Lia z, İftE [0, 1] rational; -z, if x [0,1 irrational 1 f(x) = if z E (, i] rational; Prove that the function f is not integrable on
Question 4 of the image
Prove that, for all n 1 1 Arrange the following rational numbers in increasing order: (i) x, is a rational number 61/99, 3/5, 17/30, 601/999, 599/1001. g 0 2 Find positive integers r and s such that r/s is equal to the repeating decimal (ii) 2 x5/2. Find an expression for x - 5 involving x,-5, and hence explain (without formal proof) why x, tends to a limit which is not a rational number 0.30024....