1. Suppose that an < bn for all n, an → L, and bn → M. Show that L < M.

Q: Let (A, 3) be a poset, and let BC A. Prove that B has at most one greatest lower bound.

A: Let, A,&lt;~ be a poset and B is a subset of A and a∈A. Then by the definition, a is the greatest…

Q: For two R-modules Mand N Show that Hom (M,N) any with the composition is not agroup?

A: Given that M and N are two R-modules. We have to show that HomM,N is not a group under composition.…

Q: The value(s) of x that satisfy In(x² + 1) – 2 In(x) = In(16) – In(8) are/is

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Q: Find k such that 2x = ekx for all x.

A: To find k 2x=ekx

Q: 3. Use the Racetrack Principle to show that: In x 1

A: The Racetrack Principle :-

Q: 1. For every element x in A, there is an element y in B such that (x, y) E F.

A:

Q: 1. Prove that the followingn is a theorem in SD. Av - A

A: Theorem :-

Q: Suppose that E is an extension of F and a, b E E. If a is algebraic over F of degree m, and b is…

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Q: Find a counterexample to Lie's Theorem when k is not algebraically closed and chark = 0. %3D

A: According to Lie's theorem, for any finite-dimensional linear representation of a finite-dimensional…

Q: The set of integers is not dense in cofinite topo log y of R True False O

A: The set of integers is not dense in co finite topology on ℝ.

Q: Consider the set A = {1/2^{n-1}: n\epsilon \mathbb{N}}. Show that Inf(A) = 0. WITHOUT USING LIMITS

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Q: 4. Show that in the following sense: for any ɛ > 0, there exists Ro >0 so that for any R> Ro, R rp

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Q: Let fca) satis fies Cauchy Riemann Eguation , then (f(z)" is also satis fies cauchy Riemann Eguation…

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Q: Show that if x, y,z are positive integers then: z-gcd (x,y) = gcd(zx ,zy)

A: The greatest common divisor (gcd) is the largest positive integer that divides each of the integers…

Q: 2. Let ACR. If a is a lower bound of A, then it follows that -a is an upper bound for A.

A:

Q: Let x, y, z, w ∈ Z and n be a positive integer. Suppose that x ≡n y and z ≡n w. Show that x + z ≡n y…

A: Suppose that x ≡n y and z ≡n w then, (i) we are assumed that  x-y=k1 n  and  z-w=k2 n  for some…

Q: Prove that gcd(m + 1, n + 1)|mn - 1 for all m, n E Z \ {-1}.

A: Let, gcd(m+1, n+1)=x From the properties of GCD, we can say that, x divides m+1 and n+1 Now,…

Q: Suppose that {$1,...Øn} F p and I F ø; for all i < n. Show that I F þ.

A: Proof: Suppose ϕ∈Γ∗ϕ∈Γ∗. As Γ∗Γ∗ is maximally consistent it clearly follows that there exists a…

Q: in in Signed-Rank Test a two-sided alternative, we may reject H0 in favor of H1 if either w+ or w-,…

A: Signed rank test for  two tailedHo:M≠Mowe reject Ho, if W+≤Wα and W-≤Wα

Q: 2. Prove that if 5BA+ AC A - I = A, then A must be inver

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Q: For mighe. any integer exyelicsubysoup of osder Show tht /2 has

A: Since, you have posted multiple questions but according to guidelines we will solve first question…

Q: Use Simpsons 1/3 rule to evaluate of e-x2 over the interval 0.2 to 1.5 using 6 segments ?

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Q: Show that for every x E IR, there exists an n EN such that <x<n.

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Q: (c) Show that a permutation o E S, commutes with r if and only if the permu- tation f(o) commutes…

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Q: Prop (1.23 ≤ x ≤ 1.72) = Prop (-2.50 ≤ x ≤ 0.5) = Prop (x ≥ 3.00)

A: here use normal distribution table

Q: If x is a lower bound for S in R then x must be unique..

A: If x is a lower bound for S in R then x must be unique.

Q: 1. For every element x in A, there is an element y in B such that (x, y)EF

A: Remark: Let A and B be two non-empty sets. A relation R from A into B is subset of A*B. A function f…

Q: prove for any 2 non negative integers x and y, where x>y, if their gcd IS Y, THEN THEIR LCM is x

A: The objective is to show that LCM(x,y)=x, given GCD(x,y)=y; where x and y are non-negative integers…

Q: see the equation as attached here

A: Solution:Given    0&lt;an≤bn and∑n=1∞an divergesTo prove:∑n=1∞bn diverges

Q: FOR ANY TWOCOMPLEX NUMBERSZ, ANDZ,,PROVE THAT |Z|+|Z,|=,(Z, +Z,)+ /Z,Z» +÷(Z, +Z»)- Z,Z, .

A: for two complex number Z1 and Z2 prove that Z1+Z2=12Z1+Z2+Z1Z2+12Z1+Z2-Z1Z2

Q: Prove that Theorem 13.1 (vi) using the simple ftn, nonnegative ftn and general ftn. (vi) If X is…

A: vi) From the given information, X is measurable in N.  There exists a sequence of simple functions,…

Q: 10. Prove that Z has no smallest element.

A: we will prove it by contradiction, Suppose x belongs to Z  is the smallest element.

Q: 3. Prove that Re(2-+z')<4 when z<1.

A: a.          x+y ≤x + y                                             (triangle inequality)b. Re(z1+z2)…

Q: Show that Z4 is not isomorphic to U(12) by comparing the orders of elements.

A: z4=0,1,2,3 is group w.r to +order of elementsoa=n⇒an=e where n is least positive…

Q: If tanh '() = 1- 12/3, then z = %3D In + 종 In O In In

A: As per the question we have to solve the complex equation and find the exact value of the complex…

Q: Suppose a E Z. Prove that for any n E N, 5|2"a implies 5|a.

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Q: 1. Prove that fraction where m is z-4

A: Given: zn-1z-1 ⇒zn-1nz-1 ----(1) Let apply the limit in equation (1) limz→1zn-1nz-1 ----(2) Let n=pq…

Q: Let m and n be positive integers with gcd(m,n) = 1. Prove that Z/mZ x _Z/ nZ = Z/mnZ

A: Suppose m and n are relatively prime.

Q: 3. Suppose (a,n) is Cauchy and that for all m, k 2 100, Jam - as 0. Does it follow that the…

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Q: Show that cl(a) = {a} if and only if a ∈ Z(G).

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Q: Show that For all z € C, with z + 0, then: |z|?z=1 = z

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Q: 2. Prove thatN is the smallest inductive subset of R:

A: We can prove this by proving property of induction

Q: Prove or disprove: If xn ≥ 0 for all n ∈ N and lim sup xn = 0, then xn → 0

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Q: If (an) has limit−1 and (bn) tends to infinity, does it follow that only finitely many terms of (an…

A: No it doesn't follow the statement

Q: (a) Show that for x > 0, 1 5a' < In(1+ x) < |

A: As per guidelines, we will solve only first part. Kindly repost other part separately.

Q: In Z, let H = &lt;5&gt; and K = &lt;7&gt;. Prove that Z = HK. Does Z = H X K?

A: Given: In Z H=5K=7

Q: Is f(x, y, z) = 8 In x+7 In y+5 In z homothetic? Of(x, y, z) is not homothetic Of(x, y, z) is…

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Q: prove the following:

A: Given,           f(x-y, x+y)=2(x2+y2) To prove that,          ∂f(x, y)∂x+∂f(x, y)∂y=2(x+y)

Q: By expanding both sides of the identity (1+ x)"+4 = (1+x)"(1+x)* prove that ("**) - (") + «(,",) +…

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