8. Let w, z EC such that |w| > |z|. Prove that (n+1) | ²7. < W for all n E N. |w| |w| - |z|

Q: Suppose X ~ BIN (100, 0.2). Identify the correct continuity correction for finding P (X < 21).…

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Q: - Let (fn)1 be a sequence of functions such that fnf on a set X. Which of the following condition(s…

A: X should be a closed interval.

Q: 2. Proof by Cases Suppose you are given four natural numbers w, x,y,z e N such that w > x and y > z.…

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Q: Suppose the function g is on the interval [a, b], and |g(x)|< B for all x E R. I) Prove that |g(x)?…

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

Q: Suppose h is continuous on ℝ and h(q) = 0 for every rational number q. Does it follow that h(x) = 0…

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Q: 2. Show that by using MEAN VALUE THEOREM . u-v < tan-1 v - tanu < if 1+v2 u-v 1+p2 0< u<v and ded…

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Q: 6. The Archimedean Property of the real numbers is the following statement: For every x, y ER such…

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Q: b) Let f :R R , then prove that the following three statements are equivalent. (i) f is continuous…

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Q: 12. For each n € N, define A, = [0, 1+] and B₁ = [0,1]. - =1 (a) Prove that U1A,, is an interval,…

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Q: 17. Show that f(x) = x² +2x is uniformly continuous on the interval [0, 10], using the e -8…

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Q: 3. Let f a nonnegative function defined on R. Suppose that for all n 21 you have to (r)dr <1 Prove…

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Q: Let X~ N(2, 9). Compute P(1 ≤ X < 7).

A: From the provided information, Mean (µ) = 2 Variance (σ2) = 9 X~N (2, 9)

Q: kx +5 for x ≤ 2 Let fx = = x-1 for x > 2 Find the value of k for which f(x) is continuous at x=2.…

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Q: Prove 8xyz < (x + y)(y + z)(x + z) for every x, y, z E [0, +∞[.

A: We have to prove given identity: Concept:

Q: if r<3 if x23 -m 5) Find m so that g(x) %3D mx Is continuous for all x?

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Q: Suppose you are given four natural numbers w, x, y,z e N such that w>x and y > z. Show that ly – w|…

A: It is given that w, x, y, z are natural numbers such that w&gt;x and y&gt;z. Now we want to show…

Q: 10) Define the metric d on R² by d (x₁, y₁), (x2, y2)) = max{|x₁ - x₂], ly₁ - yal}. Verify that this…

A: This is a question of topology.

Q: 2.22 Let f be continuous at a, and e > 0. Show there exists & > 0 such that I1, 12 E Df n (a -- 8, a…

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Q: 8. On which of the following intervals is (x – 1) (x + 4)(x +2) < 0? A {x|-4 < x <-2} B {xl-4,<x<–2}…

A: Given that: (x-1)2(x+4)(x+2)&lt;0

Q: 18.12. Let the function f be entire and f(z) → o as z 0o. Show that f must have at least one zero.

A: We are given that f be entire and f(z) goes to infinity as z →∞  and we  have show that f has at…

Q: Suppose xo. X1....., are distinct numbers in the interval [a,b] and f € C**[a,b). Then, for each x…

A: Suppose x0, x1, . . . xn are distinct numbers in the interval  and f∈Cn+1a, b. Then for each x in a,…

Q: t K = F(u), u algebraic over F and degree of u odd. Show that K = F(u²).

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Q: (14) Let f.g E L2[a, b]. Prove If +gli + |f – gli = 2|f3 + 2|g|3 -

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Q: 4. If E(R) exists, show that |E(R)| < E(|R]).

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Q: 15) Prove that if ffx) is continuous at x=0 then so is V f(x) provided that fix) zo for all x in the…

A: Step 1: Step 2: Step 3: Step 4:

Q: 21. Let x = R". Show that ||x||∞ ≤ ||×||2.

A: The given problem is to show the given norms identity for x belongs to Rn , ||x||infinity&lt;||x||2.

Q: 2. Let (r) be the function defined by V(x) = 1. Let 1+x. In(x) Vovo...(x) n-copies Note that…

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Q: Assume that |Xn] 0. (Hint: first, prove that P(|X|>c+1) = 0.)

A: Answer:

Q: 6. Use the definition to show: if fn →f uniformly on S and 9n → g uniformly on S, then fn +9n → f +…

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Q: Consider Z-N(0,1). Which of the following statements are true? Select all that apply. For z > 0, P(Z…

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Q: Let X ~ N(10, 16). Find a bound on p(5.5 < X < 14.5). O 0.4513 O 0.5487 0.2099 0.7901 O None

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Q: 95. Prove that ||v - w2 = ||v|2 + ||w|2 – 2v • w. 97. Prove that ||v + w|[² + |v – w|? = 2(||v||² +…

A: Note:- We’ll answer the 95th question since we answer only one question at a time. Please submit a…

Q: 4. Let X = C[0, 1] be the space of continuous real valued functions on [0, 1]. Find a subset K of X…

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Q: (9) Show that the given collection F is an open cover for S such that it does not contain a finite…

A: (9) (a) Given that S = 0, 2 and F = Un | n∈ℕ where Un = 1n, 2-1n. The objective is to show that the…

Q: 4. Prove that if (an) is increasing and an ̸= 0 for all n ∈ N then ? 1/ an is decreasing.

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Q: Let f be a differentiable real function defined in (a, b). Prove that f is convex if and only if f'…

A: A function f is convex in a domain a,b if for all 0≤λ≤1, fλx1+(1-λ)x2≤λfx1+1-λfx2  ∀x1,x2∈a,b. Also…

Q: 15 Let (x+1) {x-2}" 20 g"k) = concave upX-) on what interval(s IS (NOte: Crinen gil but aske about…

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Q: Show that if f: R→ R is continuous and f(x) = 0 for every rational number a, men f(x)= 0 for all z.

A: Thank you for the question. According to our guidelines we can answer only  first question and…

Q: Let T1, T2,, En EX and r₂ > 0, for i = 1, 2, 3,..., n. Show that N₁₁ (1) N₂ (₂) n... Nr. (n) is…

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Q: Show that if f is continuous on R, then there exists a sequence (pn) of polynomials such t Pnf…

A: To show that if f is continuous on R, then there exists a sequence (pn) of polynomials such that…