Prove For every real number m, there exists an integer a such that |m−a|<1.

Q: Prave that for all positive integers 22k-1 is multiple of 3.

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Q: 2. Show that for any real number c, there is exactly one integer in the interval (c,c + 1].

A: As per guidelines, I can do only first question.

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

A: Let m be any integer and x be any real number that is not an integer.

Q: Prove that for every real number x, 3x − 6| ≤ x iff |x − | ≤ ³1.

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Q: 1) In Exercise 4.78, it was stated for every two positive real numbers a and b that + > 2. Use this…

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: Let a,b and x be real numbers with a0. Show that there exists a rational number t such that a <

A: We know that the real numbers are densely ordered. This means that between any two distinct real…

Q: Prove that there does does not exist a unique positive real number, x, with the property that x…

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Q: Let x1, x2,...,n be real numbers. Prove that |X1 + X2 + ··· + Xn] ≤ |X1| + |X2| + · · · + |xn|·

A: The objective of this question is to prove the triangle inequality for a sequence of real numbers.…

Q: Let a be an algebraic integer in Q(v-37) and let A = (2, 1 + v-37). Prove that either a or a –- 1 is…

A: Given that,         α is an integer in Q-37 lets consider  α=-37⇒α-1=-37-1

Q: Prove that for each irrational number r, there exists a unique integer n such that |r – n| < 1/2.

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Q: Prove that for any two integers a and b with b>0, there exist unique integers q (the quotient) and r…

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Q: Let n be an integer. Apply any methods to prove that n? can be written as either 3k or 3k + 1 for…

A: By the quotient remainder theorem, any integer n can be written as n= 3q+r, that is n=3q or n= 3q+1…

Q: Which proof method is used in above proof. O Proof by counterexample O Direct proof O Proof by…

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Q: Prove that there are integers x and y such that 10x-13y=1

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Q: For every real number there exist an integer m such that m<b ylgn ihi

A: Use Archimedean property of real numbers.

Q: Prove: If m and n are non-zero integers and m > n, then there exists an integer q such that (1/n -…

A: Let m and n are non-zero integers such that m&gt;n To prove there exists an integer q such that…

Q: If a and b are positive integers such that gcd(a,b)=13, then find the sum of all possible distinct…

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Q: Prove directly that no number of the form 4a (8b − 1) can be written as the sum of three squares by…

A: Let us first show that (8b-1) cannot be expressed as a sum of three squares Let us take any integer…

Q: Let A be the subset of the real numbers given by A = {1 + a : a ∈ R and a > 0}. Show that inf A =…

A: A={1+a:a∈R and a&gt;0}Now,1+a&gt;1 for all a∈R and a&gt;0Thus 1 is a lower bound of A.We shall show…

Q: Let k and n be two positive integers, and assume that n is odd. Prove that there exists an integer a…

A: Given: Let k and n be two positive integers in that, we assume if n is odd.

Q: Prove that, for all real numbers r and all integers k and m, (O()-OC) r- k %3D

A: To prove the below statement for real  number r and all the integers k and m. rmmk=rkr−km−k

Q: a. If b is any nonzero real number, prove that |1/b| = 1/| b | . b. Prove that |a /b | = |a|/| b|…

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Q: if z1-z2 and z1/z2 are both real then prove that two must either real or equal to each other

A: Suppose z1-z2 and z1z2 are both real. We need to prove either z1 and z2 are real or equal to each…

Q: Prove by contradiction that for all integers a, b, and c, if a|b and a | c, then a |(b + c).

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Q: Let a and b be any real numbers such that a 0. Prove that a < b.

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Q: Prove that, if a, b, and c are real numbers such that a <b< c, then |b| < max{Ja|, Jel}.

A: Consider the given: a, b ,c are real numbers  such that a&lt;b&lt;c

Q: Is every infinite subset of Z x (Z-{0}) a rational number?

A: Given: Infinite subset of ℤ×ℤ-0 To Prove: Every subset is a rational number. Rational Number: Any…

Q: {x|x is a real number such that x² = : 1} {x|x is a positive integer less than 12} {x|x is the…

A: SOLUTION -(a)

Q: 7. Prove that for each real number x, there is a unique integer n such that n ≤ x ≤ n + 1.

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Q: Prove that for all real numbers x > 5 holds x+1 11/≤1²/1 2x + 3 2 Then find a real number M such…

A: Given :x+12x+3-12≤12