3. Suppose (a,n) is Cauchy and that for all m, k 2 100, Jam - as <1/4. Let e> 0. Does it follow that the interval (a100 - - e, a100 + e) contains infinitely many terms of (a,)? Either show that it does or give a counter-example.

Q: - Show that S¹ = {(x, y) = R²|x² + y² = 1} and D² = {(x, y) = R²|x² + y² ≤ 1} have the same…

A: To show that the unit circle and the closed unit disk  have the same cardinality, establish a…

Q: 6) If X is N(45,5) for i = 1, 2,..., 25; find: 25 (a) P (30 < (X-45) < 55) 221 b) P(30< (X -45) 2 <…

A: The objective of the question is to find the probability of two different scenarios for a normally…

Q: How many 5-tuples (x, y, z, u, v) of natural numbers satisfy x + y+ z+u + v = 20? Let x= X-1,…

A:

Q: 2. LetG = (Z12,) & H == {4,8,0}. Then [G:H]=3. *True *False

A:

Q: 10. Show that the interval (0, 1) is covered by the open intervals In (1/n, 1- 1/n), n 2 3, but (0,…

A:

Q: 5.3.2 (a) Suppose that n > 3. Prove that ("+1)ʻ 5.

A: Given:-             n+1n2 &lt; 2 n+1n2 -22&lt;0n+1n+2n+1n-2&lt;0n+1n+2 =…

Q: [a, b] = N=1 (a -,b+-) To prove this, show the following: То Show that [a, b](a – ÷,b +- ) for every…

A: Given that, [a,b]=∩n=1∞a-1n,b+1n This can be written as, ∩n=1∞a-1n,b+1n=[a-1,b] The aim is to show…

Q: 5.-find the exponential Ronction fa)= ab? tha satisfies the condition f(0) = -4 and f(-3) = - 500

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Q: 2 Prove that | (0,1)| = |[0, 1]|. (Hint:

A:

Q: 5. Let F= [3,2,1] and r= [2,1,–3]. Compute the following: %3D a) Proj „F. b) Orth rF = F – Proj „F.…

A: We know that  ProjrF=r.F∥r∥2r Now, r.F=23+12-31=5 ∥r∥2=r,r        =2,1,-3,2,1,-3        =22+11-3-3…

Q: 3. Is this sentence consistent? “There is someone who loves exactly those people who do not love…

A:

Q: let fAg be funcbions that sabis fy fla)= -4 A g la)=-9. find h'la) for: a) h(x) = 2f(x) c) h(x)= 10…

A: Since multiple sub parts ate asked, I am answering first 3 subparts.

Q: 17. Let P1.Pn be positive real numbers such that P1+ + Pn=1. (a) Prove the following: @P² ≥ (ii)…

A:

Q: Lise a direct proof to show "fx's odd and y is even, anen x+1s odd."

A: The given statement is: "If x is odd and y is even, then x+y is odd" Result: Any odd number x can be…

Q: Let A = {n E Z n = 5r for some integer r} and = {m e Z | m = 20s for some integer s}. Prove or…

A: We need ton prove or disprove given statement. We will prove in general and To Disprove we need only…

Q: . show that for every positive integer n, if n^2+2 is prime then n is a multuple of 3. Is the…

A:

Q: 5. Let x and y be irrational numbers such that x-y is also irrational. Define sets A and B by A = {x…

A:

Q: 1. What R code will compute P(X125)? 3. What R code will compute P(160<X<170) ?

A: X ~ N(150, 12) Mean(µ) = 150Standard deviations (σ) = 12

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.