True or False? Prove your answer Suppose A is a countably infinite set of real numbers that is bounded above, and let S = sup A. Then for every ε > 0, the interval (S − ε, S) contains a number in A and a number not in A.

Q: 2. Show that for any real number c, there is exactly one integer in the interval (c,c + 1].

A: As per guidelines, I can do only first question.

Q: Which of the following is an example of an infinite set which is bounded and countable? * A.) The…

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Q: Let A be a subset of real numbers bounded from below. Then, the set-2A = {-2x: x E A} is bounded…

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Q: Let n be an odd integer, a > 0, and b < 0. Prove that x" + ax + b= 0 has a unique root. Hint: If…

A: Let n be an odd integer, a &gt; 0, and b &lt; 0. Prove that xn + ax + b = 0 has a unique root

Q: 4. Show that Archimedean Principle is a consequence of the assertions that every interval (a, b)…

A: To prove that the Archimedean Principle is a consequence of the assertions that every interval a, b…

Q: For every real numbers a and b, b = 0, |a| |b| b

A: Here we need to show that for any real numbers a and b,b is non zero,    ab=ab. Case-1: if a=0 and…

Q: Is it possible to find a number  n  such that is ∫0∞  xn dxconvergent?

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Q: xample 1.22. Let p, q be two positiv ppose (a1, a2,..., an) is an n-tuple of n Σακοκ k=1

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Q: Find a set B that is well-defined such that 10 ɛ B and 5 ɛ B. B = {x|x ɛl and x < 11}, I the set of…

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Q: Activity 7.2. Even if we omit the negatives, the real numbers fail to satisfy the well-ordering…

A: Given that the set R+ of all positive real numbers fails to be well ordered.

Q: Find all integers n such that o(n) = 40.

A: Use the multiplicative property of ϕ:  if n and m are relatively prime then  ϕ(nm) = ϕ(n)ϕ(m). 40  =…

Q: I. Prove the following statements using the indicated proof. 1.) Let 1 € Z. If z is odd, then (1²…

A: Power of an even number is always even.

Q: True or false (PROVE YOUR ANSWER) :-If A is a set of real numbers that is bounded above and sup A∉…

A: The statement is True.   Proof: Given that A⊆ℝ is bounded above and sup A∉A Let b=sup A Then for any…

Q: Let n be a real number and let E be a positive real number. Determine whether there exists a real…

A: |1/(n+1) | &lt; E =&gt; -E &lt; 1/(n+1) &lt; E From the first inequality, we get that,  n+1 &gt;…

Q: of T2, T-:

A: Consider Tn={n, n2}, where n is an integer. The objective is to find the number of elements in each…

Q: Let R be the set of all real numbers. Determine every function g: R→ R such that for all real…

A: We need to find all functions g : ℝ→ℝ that satisfies g(a2+ b+g(b))=2b+(g(a))2.

Q: For any integer m and any real number x, if x is integer, then x]+[m-x] = m- 1. -. not an

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Q: • If f and g are completely multiplicative arithmetic functions, then f *g must also be completely…

A: If f and g are completely multiplicative arithmetic functions, then f*g must also be completely…

Q: 1=1,f2 =1 and fn+2 =fn +fn+1 for all n≥1.    Prove that for all natural numbers n≥1: f1 f2 +f2 f3…

A: As the domain for the result is natural numbers, n≥1, we can use principle of mathematical induction…

Q: 6. Find the inverse Laplace transform of the function (8² – a²)(as + b) I(s) = (s² + a?)(s² + B2)

A: Given - Let a≠0, b≠0, c≠0, α≠0, β≠0 and γ≠0 represent any real nonzero constants, such that c&gt;0,…

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.