(a) Provide an example of a sequence (an)neN such that a2m → 2 but (an)nen does not converge to 2. (b) Draw the graph of a function ƒ : [0, 1] → R, with ƒ(0) < 0, ƒ(1) > 0 and ƒ(x) ‡ 0 for all x = [0, 1].

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

A: Step 1:

Q: True or False? Prove your answer! If f : R → R is continuous and (xn) is a sequence such that…

A: If f : R → R is continuous and (xn) is a sequence such that (f(xn)) converges to f(c), then (xn)…

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

A:

Q: Assume you have given a sequence c,, with non-zero values (cn #0 for n = 1,2,...) that fulfils the…

A:

Q: (a) Show that the sequence is strictly decreasing and downward bounded. (b) Show that the sequence…

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Q: (a) State (without proof) the extreme value theorem. (b) Give an example of a function ƒ : [0, 1] →…

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Q: (a) Show that the sequence is strongly decreasing and downward bounded. (b) Show that the sequence…

A: The objective of this question is to prove that the given sequence is strongly decreasing and…

Q: (a) Prove that (2x + 2) > 1 for all x > 0. (b) Let so > 0, and sn+1= } (2s, + ) for all n > 0. Prove…

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Q: (c) Let (sn) be a sequence of negative numbers (sn <0 for all n E N). Prove that lim 8, = -00 if and…

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Q: 3. Assume that f : R → R is such that |f(x) – f(y)| 0. (a) Prove that |æn+1 – Xn] 1. (b) Prove…

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Q: (c) A non-constant function f : [0, 1] → R such that f(x) > 0 for any x e [0, 1]I f(x)dx = 0. Give a…

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Q: 3. A particle moving along the x-axis has velocity function v(t) = t²e-t. How far does the particle…

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Q: For each of (a)-(d) give a sequence (rn)1 with the stated prop- erties. In each case, the sequence…

A: As per our guidelines we are suppose to answer only 3 sub-parts.

Q: 1. Recall the definition of the limit of a sequence of real numbers (xn) = (x1, X2, X3, . . .) "(Tn)…

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Q: Suppose that the function f:R → R be differentiable and that {xn}is a strictly increasing bounded…

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Q: Let (gn) be a sequence of continuous functions that convergesuniformly to g on a compact set K. If…

A: Given: Let (gn) be a sequence of continuous functions that converges uniformly to g on a compact set…

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: Let (Jn)neN be a sequence of functions f: R → R satisfying |fn+1(c) – fn(c)| 2 and neN , and ae(0,…

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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: Let ƒ : [−1,1] → R be a continuous function. Prove that the sequence (n) defined inductively by Xn 1…

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