Theorem 24. If ɛ > 0, then there is an integer N EN such that 0 < + < e. .

Q: )"-( 18 6. lim V n4n +(4n)" 2+ n2 4 4+ n2

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Q: VI+ rdx < (2e – 1)

A: Integration in calculus is just the opposite of differentiation. We differentiate functions to get…

Q: 6. Let a be a number written (in base 10) as ao 10⁰+ a₁ 10¹ + a₂. 10²+...+an. 10" where 0 ≤a < 10.…

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Q: 10. Prove that (n) ≤ n/2 for n ≥ 8.

A: To prove that π(n) ≤ n/2 for n ≥ 8, where π(n) represents the number of prime numbers less than or…

Q: I need help with this problem and an explanation for the solution described below (Discrete…

A: The objective of the question is to find the negation of the given statement. In logic, the negation…

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: 2 4) The graphs of y=x and y=x make it obvious that x >x for all real number x > 1. Use induction…

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Q: (a) Show that if n € N, then n+1 In dx. I

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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,…

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Q: 5. Let r > 1 be an integer such that 2" + 1 is prime. Show that r = 2" for some non-negative integer…

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Q: Find the supremum and infimum of the set x € R:x = +(-1)" ,n e N} + (-1)" ,n e N}

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Q: 3.5. Let n E N. Prove that if n + ÷ < 2, then n² + n2 <4.

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Q: 19. Let s andt be real numbers such that t - s > 1. Prove that there exists an integer p such that s…

A: Here we use the concept of greatest integer function. For a real number X, [X] denotes the largest…

Q: 5.80 Show that these facts are true. Use the definition of binomial coefficients to show that each…

A: We expand left hand side i.e. nCn nCn= n!/n!*(n-n)! We know that 0!= 1 so  nCn = n!/n! = 1 This…

Q: Prove that 6 divides n³ - n whenever n is a nonnegative integer.

A: In this problem we have to prove that  divides  where  is a nonnegative integer.

Q: Show that for integers a, n satisfying 0 < a <n, ged(a, n) = gcd(n, n-a)

A: Solution is given below:

Q: Prove that for all n € N, 91(4" + 6n + 8) and n € N, 31(n³-n).

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Q: Consider certain real numbers ui and vi, i = 1,..., n, n <∞0. Show that (u₁v₁ + U₂v₂)² < (u² + u²)…

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Q: Assume that 0  ≤ a1 ≤ 1 and that  an+1=3√(1+an)/2 for all n ≥ 1. Prove that (a)0  ≤ an ≤ 1 for all…

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Q: Let F(n) denote the Fibonacci numbers. Prove the following property for n ≥ 3 directly from the…

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Q: Compute the infimum and supremum of the set {n+(1/n)" : n E N} if they exist.

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Q: n² < 2" for n = 4,5, 6, ... (Hint: 2n + 1 < 2" for n = 3,4, 5, ...)

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Q: Prove that there exists an n ∈ N such that 7 | (3^n − 1).

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Q: Prove that xk = x* mod |G| for x € G for any integer k.

A: Prove it considering two cases.

Q: How do I prove the following statements: (a) For all a, b, c ∈ Z, if a a. ● (O2) For all a, b ∈…

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

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Q: 4. Prove that if (an) is increasing and an ̸= 0 for all n ∈ N then ? 1/ an is decreasing.

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Q: 6. Prove that 3" < n! whenever n is a positive integer greater than 6.

A: We need to prove that ,  3n &lt; n!  for every positive integer n greater than 6. We will use method…

Q: n + 5 > 4 n

A: We know, the square of an integer is always non-negative (n - 2)² ≥ 0 for all integers n

Q: (14) Let E denote the set of even integers, O denote the set of odd integers, and x, y € Z be any…

A: I have given the answer in the next step. Hope you understand that

Q: Prove that for every positive integer n the number (1 – /2)" is not ratioanal.

A: Given: The given number is, (1-2)n where n is the positive integer.   To prove: The number (1-2)n is…

Q: prove if limit n -&gt; infinity xn = x and if x &gt; q then there exists k belonging to the natural…

A: Step 1: Step 2: Step 3: Step 4: