Challenge. Establish the Cauchy-Schwartz inequality: For real numbers we have (a + az + ..+ a) (b+ b3 + .+ b) > (a1b1 + azb2 + .. + anbn)? with equality only if there is a constant k so that a; = kb; for each i.

Q: Let the universal set be the set R of all real numbers and let A={x€R|-2<x<2} and B= {xeR|-3<x<0}…

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Q: Prove that for all positive integers n, 1 3 2n – 1 1 X...X 4 V3n 2n HINT: You will find it difficult…

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Q: Find the maximum value of a, x, +a,x, + .+ax to the condition that x +x, +. + x =1 for given…

A: We need to find the maximum value of a1x1+a2x2+a3x3+...+anxn to the condition that…

Q: 18. How many integer solutions are there to the inequality x1 + x2 + x3 + x4 + X5 ≤ 782 provided…

A: We have to find the number of integer solution for

Q: Let a, b, c be three consecutive odd nubmers, with a < b < c. What is the  value of a2−2b2+c2?

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Q: If a = 2a - a² and b =1-(1-a), show that a = b for all natural numbers n. Assume that a, is a fixed…

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Q: 4. Show that Archimedean Principle is a consequence of the assertions that every interval (a, b)…

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Q: Assume the following three statements are true. p: For all real numbers x and y, if xy is…

A: We are given that statements p,  q and r are true. We need to prove that 2a-1 or a2+3 is…

Q: Number 2 only. Thanks!

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Q: ind the interval in the real number line which equals 1 U [4+, 20- n n=1 n

A: Answer given below.

Q: 2. Definition. Let r be a non-negative positive real numbel A square root of r is the Iinique…

A: Note: Square of any real number must be non-negative and addition or subtraction of any two real…

Q: 2. Prove the statement: The sum of the squares of any two consecutive integers is odd.

A: I have assumed that n be any integer. Then i have found the sum of square of two consecutive…

Q: In = 5n+3 2n+1

A: xn= 5n+32n+1Therefore limn→∞5n+32n+1= lim5+3n2+1n=5+02+0=52n→∞Therefore this is a convergent…

Q: 5 Find the least common denominator of X 11 and 4 8x+5' X 3

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