1) Prove both versions of the triangle inequality: for all a, b = R", we have ||a + b|| ≤ ||a|| + || b || and ||a|| – ||b|| ≤ ||a – b||.

Q: 3. Consider the Cauchy-Schwarz inequality. That is, for u, v E R": Use some reasoning, or do some…

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Q: Prove: | a | - | b | <= | a + b |

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Q: 2. Find p and q such that 7x+1 5 |5x-1|≤4 implies p ≤: <q

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Q: 6. Let a, x, and y be real numbers and let e > 0. Suppose that |x-a| < e and |y-a| < e. Use the…

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Q: Prove that for any sets A, B, X - X − (An B) = (X − A) U (X – B). -

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Q: Prove that for any x, y ≥ 0, we have √x - √y ≤ √√x - y). (Hint. Consider the cases x≤y and y ≤ x…

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Q: 4) Let a < b be real numbers. Prove that the half open/half closed interval (a, b) is both an Fo and…

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Q: Let a and b be real numbers. (a) Prove that a| – 16| < la + b| and |b|- lal < la + b|. (b) Use part…

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Q: (c) Let z1 = 2+ 2i and let 22 =1+i. Prove that the triangle inequality |21 + z2| < |21| + |22|…

A: Left hand side is equal to, z1+z2=2+2i+1+i=3+3i=32+32=32 Right hand side is equal to,…

Q: 5) Suppose that W1 < R" and W2 4 R". Prove (W1 n W2) < R"

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Q: Prove if this is true of false. We know |x| = x if x ≥ 0, and |x| = -x, if x < 0. Then, for all…

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Q: Prove that |A U B| + |A n B| = |A| + |B|

A: Let A and B are two sets. A = number of elements in the set A. B = number of elements in the set B.…

Q: Prove that −3 ≤ x ≤ 2 implies that |x² + x − 6| ≤ 6|x − 2|.

A: We have to prove thatWe know that

Q: 3. Prove the triangle inequality: For any real numbers x, y, |x + y ≤ |x| + |y].

A: For any real number  we have to prove

Q: Let a be any number from R, and let b be a number from R that is positive. The solution interval to…

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Q: a) Prove that |x| ≤ 2 implies that |x² − 4| ≤ 4|x − 21.

A: To prove the given statement, let's break it down into a few steps: Starting with the assumption…

Q: . Consider the statement: Let A, B ∈ Z. If 6|A and 10|B, then 4|(A2 + B2). (a) Prove the statement.…

A: 5. Consider the statement: Let A, B ∈ Z. If 6|A and 10|B, then 4|(A^2 + B^2). (a) Prove the…

Q: Which of the following defines the domain of |4x-8<12? A) -8<x<-4 B)-4 <x < 8 C) -1<x< 5 D) 1<x<5

A: We can solve the inequality in order to get the solution.

Q: Let a,b be two different points in R^n one. We notice that d=(a,b)>0 (why is that so ?). Prove that…

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Q: If x and y are positive real numbers, use the direct method to prove the following inequality: 7 5…

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Q: Whenever one has an inequality, an interesting problem is to characterize the cases of equality.…

A: Suppose, x ≠ 0, y ≠ 0 , i.e., both x and y are non-zero and let, x and y are of different signs.…

Q: Let H = Span{1+x,1+x², 3+x +2x², 2 + x + x²}. what is dimH?

A: Solution

Q: Using only the Axioms and Elementary Properties of the real numbers, prove Cauchy's Inequality: for…

A: xy≤12x2+y2 For all real numbers x,y.

Q: Consider the following proposed proof of the triangle inequality (different from the proof in the…

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Q: 2. Define the absolute value of x E R. then find all real numbers x such that |x + 1 < |x – 1).

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Q: Let a, x, and y be real numbers and let e > 0. Suppose that |x-al < e and y-al <e Use the Triangle…

A: Given that, a, x, and y be real numbers and let ε &gt; 0Also,x-a&lt;εand y-a&lt;εWe know…

Q: I have the following triangle inequality proof, I just dont understand any part of it so if able…

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