Let En 0 and (yn) be bounded. Prove nyn → 0.

Q: 2. N₁₁ (1) Let x1, x2,..., n E X and ri> 0, for i=1,2,3,..., n. Show that Nr₂ (₂) n... Nr (n) is…

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Q: Suppose X and Y are r.v.s such that E(X) = 4, Var(X) = 48, E(Y ) = 2, andVar(Y ) = 5. Find upper and…

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Q: R (T) is bounded by its value at the origin i.e., IR, (t)l<Rxx (0) XX XX

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Q: 4. Let X₁, X2,..., XnEX and r> 0, for i = 1,2,3,...,n. Show that U=1 N (X) is bounded.

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Q: x*y´+ xy´e xy = o %3D find all values of x fon wich all. the soluhons of thu's are bounded when x…

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Q: S (-1,3] U [5, 6) Ø {x ln(x) x>0} inf (S) sup(S)

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Q: Let h: R→ R be continuous on R satisfying h(m/2") = 0 for all me Z, ne N. Show that h(x) = 0 for all…

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Q: Is the following bounded or unbounded and why? f(x,y) = ln(x2 + y2)

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Q: Let Xi ∼ Bernoulli(1/2), and let Yn = X1 + X2 + · · · + Xn. Use the central limit theorem to find…

A: Xi ~Ber(1/2) ; i=1,2,...,n

Q: Q. No. 2 Let N be the set of natural numbers and N = N U {∞}. Define a function d: N* × N* → R as…

A: I have proved the three conditions for metric

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A: Given -  To Prove -

Q: 4. Let f(z) = cosh z. (3) Show that f: S→ f(S) is ono-to-one.

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Q: 14k 14 as a definite integral. 8 Express the limit lim Σ n→00k =1 9 + + 121

A: Known fact: Let f be a continuous function on a,b. Then ∫abfx dx=limn→∞ ∑i=1n fxi ∆x where ∆x=b-an…

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Q: Prove thait if y=f(x) f(x)=0 when x is a rotional number, then f(x)=0 for all x € R. is continuous…

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Q: -Prove that 2x² - 8x = 22 has at least one soution on the interval IVT. [5,10] by using

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Q: 7.2 Give an & - 8 pr at (1 continuous (2x - 2).

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Q: 6. Let f: Q R be defined by f(x) Show that f is continuous. 0, :{ 1, = if x € (-∞, √2) nQ if x €…

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Q: Suppose X~U(a, b), with µ = 1 and ² =. Find P[X < 0]

A: Solution-: Suppose, X→U(a,b), with μ=1  and  σ2=4/3 We want to find P(X&lt;0)=?

Q: Queston #o 4 Show that X, Ixe, where x, and Xe E IR X= (1,1,1, -1) Xz = (4,2,2, -1) %3D

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Q: Suppose that fand gare continuous functions such that f(z)dz = 2and s(z)dz = 3, find (2f(2) +…

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Q: Show that a continuous function f : [0, 1] → Z is a constant function.

A: Consider a function .We need to show that a continuous function  is a constant function.We know…

Q: el 29-38 Determine the set of points at which the function is continuous. 29. F(x, y) = xy 1 + e*- 1…

A: Solution:

Q: 6. Let f, g, h : [a, b] → R satisfy f(x) < g(x) < h(x) for all x E [a, b]. Suppose that f and h are…

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Q: 5. Let f: Z→ Z be defined by f(x) = 2x-1. Determine whether f(x) is onto and/or 1 to 1. Explain your…

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Q: 5. For f(x) = x lnx, x € [1,3], find the least square approximation of f in the space P₂.

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Q: = [[ F - a5, I = S F = î4æz – jy² + kyz 1

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Q: 2. Let function f [0, 1] → R defined by f(x) = integrable on [0, 1]. Evaluate f f(x) dr. 659 0, if a…

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Q: 13. Let f(z) = z* - a* where f (z) e ¢. Factorize f(z) completely.

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Q: If n is a positive even integer and f(x) = Vx, then f is continuous everywhere. True False

A: Given that n is a positive integer and f(x)=(x)^1/n.

Q: Suppose that f : R → [0, 0) is measurable. (positive) show that if f R then F=0 a.e

A: For the solution follow the next step.

Q: that f 1s a continuous on IR Creal a nd h ale odif feren tiable funcho ns mberr). I denhfy a for…

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Q: If Z = f(x + g(y)) , prove: Zx * Zyx = Zy * Zxx

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Q: f(x + a) dx = f(1) dæ = | f(x) d.r. -T+a

A: Proving of integral of periodic function

Q: Show that if f is Riemann integrable [a, b] and f(r) 0 for all r € [a, b], on f(x) dr 2 0.

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Q: Z = In z + X + find the det ermain Z as Function given equation adiff erential X änd

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