Let f: A R and f is continuous at some cluster point c EA. If (x,) and (y,) are two sequences in A converging to C, such that f(x,) f(y,) for all n EA, which of the following sequences diverge: O a. All the given sequences converge. Ob. f(x,)-f,) +1 O c. (f(x,)-f(y,)) O d. (f(x,)-f(yn))

Q: 12.Q. Let 0 < aı < bị and define az = Va,b , b2 Vanba , bat1 (a, + bi)/2, ..., Ant1 = (an + bn)/2,…

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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: = 5. Prove that if x1 = 1 and xn+1 = V6 + xn, then xn 1. e 6. Let E CR such that E is nonempty and…

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: (f) If (a„) is convergent, then (a,) is monotone and bounded. (g) If f is continuous and A is open,…

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Q: 2. Construct a sequence of unbounded functions In : [0,1] → R that converges pointwise to a bounded…

A: Step 1:

Q: Proposition 4.6.10. Let f : R → R be defined by f(x) = +b where a andbare constants. Then f is = ax…

A: Given: fx=ax+b Let c be any arbitrary point in ℝand xn:n∈ℕ be a sequence in ℝ such that xn→c So,…

Q: 7. Suppose A and B are compact subsets of R and that fn is a sequence of functions converging…

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Q: Assume A is a bounded, non-empty subset of R. Let r = sup(A). Prove there is a sequence from A…

A: Given r=sup(A). Let r∈A.  Consider the sequence {xn} such that xn=r ∀n∈ℕ.Then {xn} is a sequence…

Q: → R is continuous and let ro E R. Prove that {x: f(x) = i

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Q: 1 Let {fn} bes sequence of real nonnegative functions on R', consider the following four statements:…

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Q: Let {ai} be a sequence in a metric space X,d. Prove that lim an = p if and only if lim d(an, p) =…

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Q: (1) Let f be a function from A to B, {An} a sequence of subsets of A, and {B} a sequence of subsets…

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Q: 3. Let e be a cluster point of a subset A of R and letf: A+R be a function. Suppose for every…

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Q: 31. Suppose f: [a, b] → R and g: [a, b] → R are both continuous. Let T = {x: f(x) = g(x)} Prove that…

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Q: 3.5.4) Show directly from the definition that of (xn) and (yn) Cauchy sequences, then cxntyn) and…

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Q: True or false: a function f: R → R is continuous at a point a ER if and only if there exists a…

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Q: . If x, y E R with x < y, prove that x < tx + (1 – t)y < y for all t e (0,1).

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: Let f be a continuous, increasing function on [a, b] and let P be a partition -a for all i = 1, 2,…

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Q: 1* Let X be the space of sequences of real numbers with only finitely many nonzero terms, with the…

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Q: 1.57 Let r, and yn be any two bounded sequences of real numbers. Prove that lim sup(xn + Yn) < lim…

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Q: 5. (A Comparison Principle) Assume that a € (0, ∞) and 3 € R. Let {n}n=0; {Yn}n=0₂ and {n}ao be…

A: The given sequences are, x0≤y0≤z0 and  xn+1≤αxn+βyn+1=αyn+βzn+1≥αzn+β,  n=0,1,2,... To Show:…

Q: Assume f: R→→ R. and for some c < 1 we have for all x, y € R, |f(x) = f(y)| ≤ c|x − y|. (a) Prove…

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Q: 2) Prove that rxz(1) <rzz(0), where r denotes the autocorrelation function of a sequence r(n) and I…

A: Autorrelation function is mathematically given by, rxx(l) = E[X(t+l).X(t)] , for all t. Similarly,…

Q: (b) Consider the set U = {(x, y) E R² : 2x + y < 1, x 2 0, y 2 0} and let Zn = (Xn, yn), where Xn =…

A: NOTE:- I HAVE REQUIRED QUESTION A TO SOLVE QUESTION C AND D. THEREFORE I AM SOLVING ONLY QUESTION B.

Q: Assume f: [0, 1] → [0, 1] is continuous. Prove there is an x = [0, 1] such that f(x) = x

A: Introduction:  This is the application of intermediate value theorem. So, we use the intermediate…

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.