Theorem 2.13: Let {an} be a sequence. Then the number L is a finite subsequential limit of {an} if and only if L satisfies either of the following conditions: (i) There are infinitely many terms of {an} that equal L. (ii) L is a limit point of the set consisting of the terms of {an}.

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Q: coprime

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Q: 8. √(T-2) 0 - x) сos лx dx

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Q: . Problem 4: critize the following proof – find all mistakes(there may be more than one). A mistake…

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A: given matrix  A=1a12a1301a23001 and I=100010001 prove  that A-1=I-N+N2-N3 where N=A-I

Q: 1. Let a be a positive real number and n be a natural number. Define S = {x E R: x >0 and " a?…

A: Note: According to our guideline we will answer the first three subparts of the question as the…

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A: This is a question of real analysis. L.H.S is a geometric series with common ratio (-x)

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Q: Prove that π2 -1 is algebraic over Q(π3).

A: We have to prove that π2-1 is algebraic over Qπ3.

Q: 3. Prove or disprove: (a) a bc = a | b (b) a | bc and gcd(a, c) = 1 =→ a |b (Hint: recall Bezout's…

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Q: verify that det(A B) = det(B A) and determine whether the equality det(A + B) = det(A) + det(B)…

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Q: Prove that if C has infimum 2 then D = {x ∈ R | c ∈ C, x = 3c} has infimum 6

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Q: Example 1.41. Both the set Q of rational mumbers and the set R - Q of irrational numbers are dense…

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Q: Theorem 1.22 If a e R. be R and a <b then there is a rational number r such that a <r< b.

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Q: 1. Show that for all z1, Z2 E C, Then we have: |Z1 + z2| < ]z1]+ |z2| 2. Show that for all z1, Z2 E…

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Q: 1 real number such that a+ E Z. Prove that a" + 1 E Z for any n E N. Let a be any

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Q: Prove that π2 - 1 is algebraic over Q(π3).

A: In this question, we have to show that π2-1 is algebraic over ℚπ3. We know that any element α in…