If x and y are real numbers and x>1, prove that there exists a positive integer n such that x" > y

Q: Prove that if a and b are integers with a 0 and where + b - a = 0, x is a positive integer such that…

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Q: 1. Let y be a positive real number. Prove that for every n e N there exists a unique positive real…

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Q: Prove by contradiction that for any positive two real numbers, x and y, if x·y ≥ 100 then either x…

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Q: Find the least integer n such that (2x^3+x^2logx) is O(x^n).

A: The given expression is 2x3+x2logx. Concept used: Let f and g from the set of integers or the set of…

Q: 5**) (a) Prove that for all positive real numbers n, r, and s, if rs < n, then r < Vn or s < Vn.

A: As per Bartleby's answering policy, we can answer only one question, so kindly post the next…

Q: Prove that if x is a positive real number, then LV L]] = [V]

A: We have to prove the given equation for all positive real numbers x .

Q: Prove. If x is a negative real number, there exists an integer k such that k >= x.

A: x is a negative real number then  x &lt; 0 So that desired integer k = 0

Q: Suppose x and y are any integers. Prove that if x and y are odd, then x + 5y is even

A: Suppose x and y are odd integers. To show x+5y2 is even. Note: Square of an odd number is odd.…

Q: Suppose x and y are any integers. Prove that if x and y are odd, then x+ 5y² is even

A: Suppose x and y are integers. Prove that if x and y are odd, then x+5y2 is even.

Q: For all integers x and y, if 6x+y is odd, then x and y are both odd. True False

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Q: If r and s are real numbers and r < s, then there exists a positive integer t such that r+t = s.…

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Q: Let a be a nonzero real number. Prove that aman = am+n for all integers m, n &gt; 0.

A: Let a be a non zero real number , where m and n are integers…

Q: Prove that for all positive real a we have 22 +5 20x4 2 4/5. 24 x2 +5

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Q: Prove that for every real number x, if x > 1 then there is a real number y such that y+ý = 2x.

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Q: prove that there exists a unique positive real number,x, with the property that x2=x.

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Q: 10. Let x be a real number. Prove that at least one of the numbers /2 – x or /2 +x is irrational.

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Q: Use proof by contradiction to show that there is no positive real number x such that (x-2)/x ≥…

A: In an inequality, if a positive number is multiplied then there is no change of sign of inequality.…

Q: Prove that there exists irrational numbers r and y such that x" is

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Q: Given x > 0 and a real number y, there is an n in N such that nx > y.

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Q: Let a and b be positive integers such that ax + by = 7 for some integers x and y. Then %3D

A: We have to choose  current answer of given statement.

Q: Let a be a positive real number and let n be an integer greater than 1.Prove or disprove that…

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Q: Prove that if m is a positive integer and r is a real number, then Lma) = l=] + -+ + + [-+ + r+ m

A: We are given a Inequality involving the sum of floor functions, and we have to prove that.

Q: Provide either a proof or a counterexample for  For all positive real numbers x, x^2 - x ≥ 0

A: According to the given information, it is required to prove the given equality or give a counter…

Q: If α is even, prove that α-1 is even. If α is odd, prove that α-1 is odd.

A: We know that the multiplication of 2-cycles can help us find any permutation of a finite set, so we…

Q: Show that for every real number ɛ 8 > 3ɛ. greater than zero, there exist a real number 8 such that

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Q: Prove that if x is an even integer, thenx^2−6x+ 3is odd

A: Any integer of the form 2m+1 is odd.

Q: . Use direct proof to prove that, if x is a positive odd integer, then z – 1 is wholly dividable у…

A: x^2-1 is divisible by 4.

Q: Let x and y be positive real numbers. Prove that a + y > 2/xy.

A: See solution below

Q: x and y be two real numbers such that x + y is rational.

A: Given statement:   x and y be two real numbers such that x+y is rational. To Prove:  If x is…

Q: For any integer m and any real number x, if x is integer, then x]+[m-x] = m- 1. -. not an

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Q: 4. Let R be the set of all real numbers, xER and n EN such that n > 2. If (1+x) >0, prove that Pn :…

A: Mathematical induction.

Q: Let a1, a2, a3, . .., an be real numbers. Prove that at least one of them is greater than the…

A: We prove it by contradiction.

Q: Find a number z that satisfies: |z| - z = 2 + i

A: Complex number which satisfy |z|-z= 2+i

Q: o Prove that for each real number x, (x + v2) is irrational or is irrational.

A: 8(a) Given x is real number. We know that 2 is a rational number. First we have to show that x+2 is…

Q: For every real number x+1, n-1 1+x+x² +x' +. x´+x' +...+x" for all neN. 1-x

A: The question has a mistake , as in the R.H.S the power of x must be (n+1).

Q: Prove: There exists a positive real number z such that z2=2.

A: Any complex number can be considered as z=x+iy where x,y∈ℝ . Note some result : If z is real number…

Q: Suppose A and B are bounded and nonempty subsets of real numbers and 1 E R. In this case: sup(AB) =…

A: Suppose A and B are bounded and nonempty subsets of real numbers and λ∈R. We need to prove that…

Q: 3. Suppose a is an irrational positive real number. Prove that if x and y are integers and x # y…

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Q: Suppose that x is a negative real number and that x < 1/x. Prove that x <-1.

A: Since we know for any real number a,b  If a &lt; b then -b &gt; -a

Q: Show that if f1(x) is O(g1(x)), f2(x) is O(92(x)), and f2(x) # 0 and g2(x) # 0 for all real numbers…

A: Big theta notation is nothing but an order notation that is used to compare the the growth rates or…

Q: Let n and r be non-negative integers such that r <n. Then, C + "C , = **C,

A: nCr=n!r!(n-r)!

Q: Show that |x^n| = |x| ^n for every positive integer n and every real number x.

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Q: Prove that there are as many positive real numbers as there are real numbers.

A: Here , we have prove that all positive numbers are real numbers.

Q: Use the Intermediate Value Theorem to prove that for every real x > 0 and every integer n > 0, there…

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Q: Prove that if æ is a positive real number, then

A: To prove that if x is a positive real number, then                               x=x

Q: Prove that for all real numbers x, if x is not an integer, then [r – 1] = [x] – 2.

A: We have to prove x−1=x−2 for all real number x, if x is not an integer. Use definition:- x is the…

Q: 11. Let y be a positive real number. Prove that for every ne N there exists a unique positive real…

A: This is a problem of real analysis.

Q: Use a direct proof to show that if x is a real number such that (x – 1)? = 0, then x³ –1 = 0.

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Q: Show that for all positive integers n and k, (~^)= (-1)^(" + k-1). k

A: The provided information is  (  k-n) Then we prove this (   k-n)=(-n)*(-n-1)*(-n-2)*.....(-n-k+1)k!…

Q: 3. Show that for any real number x, if x2 is odd, then x is odd.

A: