1 3. Let A = {= n - 2 · | n ≤ N, m € N}. m (a) Prove that sup A = 1. (b) Is 1 € A? (c) Does A have an infimum? If so, what is it?

Q: Let = e²i/n be an n-th root of unity. Show that (k(1-n) Resck (27²1) n for all 0 <k<n-1. =

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Q: 1. Express the sum 2.019+2.020 into a fraction of two natural numbers. (2.019 = 2.019191919.. .) 2.…

A: As per the guidelines we can solve 1 problem only.

Q: (∀x)(x < 1 → x2 < x). (For all positive real numbers less than 1, squaring that number actually…

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Q: p(x) = a + ax + a,x² + ... + 10x?

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Q: 3. Prove or disprove the given statement. (a) If x e R, and x + 0, then 2x > x. (b) If n e Z*, then…

A: Given: (a) If x∈R, and x≠0,then 2x&gt;x (b) If n∈Z+, then 2n-1&gt;0

Q: ·C) - G) + G)-G) + +-1 - Prove that for any positive integer n, 0.

A: Use simple property of combination.

Q: For 1 ≤p ≤ ∞, find the closer of Coo in P. (Hint: It may be practical to separate into two cases, 1…

A: We have to find the closer of C00 in lp

Q: Euler's Tution function: (n) is the number of positive integers less than n that are relatively…

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Q: c" is (n) for any real constant c> 1 and k > 1

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Q: Prove in detail that if A has infimum 2 then B={x∈R|∃a∈A, x=3a} has infimum 6.

A: Given that, The set A has Infimum 2.  B={x∈R|∃a∈A, x=3a} We have to show that B has infimum 6.

Q: Express in lowest terms. x+ x* x - x+ 1 ) 1+ x? O x(x + 1)

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Q: 1. Let a be a positive real number and n be a natural number. Define S = {x €R: x >0 and x" a?…

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Q: Let A = 21. (a) Compute A², A³, and A4. 02 (b) Find a formula for A" for any integer n ≥ 1?

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Q: Find the least positive integer n such that x¹/(x² + 1) is O(x¹), and then show that it is O(x") for…

A: Given that the function is To find the least positive integers n such that the given function is and…

Q: 3. For n e N, we define n! = (n)(n – 1)(n – 2)... (2)(1). E.g., 4! = (4)(3)(2)(1) = 24. Prove the…

A: Here we have to prove two statements.

Q: (d) Still assuming the parallelogram law, show that for any a a € R, ||ax + y||²||ax - y||² = a(||x…

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Q: If n > 1 is a composite number, then σ(n) > n+sqrt(n).

A: ⚫ For a given positive integer n, σ(n) denotes the sum of positive divisors of n. For example ,…

Q: Show that for f(n) = n³ - (ri'log, n) + 2n E O(n³ ) A give the constants: (c, n) B:arguing that your…

A: When a function is bounded above by another function with a factor asymptotically, it is referred to…

Q: 9. Prove or disprove. (a) Vn e Z*, n² < 2". (b) Vm e Z, if m is odd, then m² is odd.

A: In this question we have prove if this equation are valid or not.

Q: Use Bernoulli's inequality to prove that n + √)" = 1+ > √n for all n EN.

A: From Bernoulli inequality we know that  if x&gt;-1 then : 1+xn≥1+nx for n≤0 or n≥1. Therefore for…