1 3. Let A = {= n - 2 · | n ≤ N, m € N}. m (a) Prove that sup A = 1. (b) Is 1 € A? (c) Does A have an infimum? If so, what is it?

Q: a – b -| < 1, if|a| < 1, and |b| < 1. 1- āb - Show that|-

A: From the given information. For , then show that .

Q: is continuous at x=3, f (-1) =6 ? + a, x # 3 3b, x 3 ontinuous at x=2? - x > 0

A:

Q: 5. (a) If r23 = 0, prove that R₁ (23) = (23) = 712 + 7³1 01-23 = (1 - 12 - 13). 07.

A:

Q: (4) State which of the following statements are true, vacuously true, or false. (a) (A-B)' = A' -…

A:

Q: Q1. How many pairs of ordered numbers exist such that their sum, product, and quotient are the same?

A: Keeping a note, according to our guidelines we can only answer one question, kindly repost other…

Q: 3. Prove or disprove the given statement. (a) If x e R, and x + 0, then 2x > x. (b) If n e Z*, then…

A: Given: (a) If x∈R, and x≠0,then 2x&gt;x (b) If n∈Z+, then 2n-1&gt;0

Q: 10) Define the metric d on R² by d (x₁, y₁), (x2, y2)) = max{|x₁ - x₂], ly₁ - yal}. Verify that this…

A: This is a question of topology.

Q: ·C) - G) + G)-G) + +-1 - Prove that for any positive integer n, 0.

A: Use simple property of combination.

Q: if |z² + 1| = 2| z+1|  z e C then | z|  <= sqrt*7 Prove.

A: Given  z2+1=2z+1 , z∈ℂ            (i) Now,   z2+1=2z+1z2+12=4z+12 ,…

Q: (a) [0, 1] (c) {2,7} (e) {÷ :n e N} (g) [0, 1] U [2, 3] (b) (0, 1) (d) {T,e} (f) {0}

A: To find supremum of given sets .

Q: For 1 ≤p ≤ ∞, find the closer of Coo in P. (Hint: It may be practical to separate into two cases, 1…

A: We have to find the closer of C00 in lp

Q: この Σ * 00 L山 Refer to functions m and n. Find the function (m ° n) (x) and write the domain in…

A:

Q: 8. Find a number o such that if 0 < x − 2| < 8,then | 3x − 1| < &,where s=0.06.

A:

Q: Euler's Tution function: (n) is the number of positive integers less than n that are relatively…

A:

Q: 4. x- 1 < |x - 2|

A: "As per our guidelines we are allowed to solve only first complete question kindly repost others."…

Q: 21. Let x = R". Show that ||x||∞ ≤ ||×||2.

A: The given problem is to show the given norms identity for x belongs to Rn , ||x||infinity&lt;||x||2.

Q: 3. For n e N, we define n! = (n)(n – 1)(n – 2)... (2)(1). E.g., 4! = (4)(3)(2)(1) = 24. Prove the…

A: Here we have to prove two statements.

Q: a²-a² = [2²2² (2²]. Let A = Then for any a, Select one: det(A) ≥ 0 O A is singular O None of them O…

A:

Q: Solve for the moment generating function of X if for all natural number n, X ∼ U {1,2,…,n}.

A: Given that, X ∼ U {1,2,…,n} , we need to solve for the moment generating function of X if for all…

Q: (2 Let xinN(0.1), n = 2 and T= Xx +X2 Show that Tis a Sufficiency Stakistic.

A:

Q: 3. Let r = (4, –3, 1, –5) € R'. Which of the following equals ||r||? (a) 5 (b) 13 (c) V51 (d) v63

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Q: Let x, y E N, determine each of following statemen is true or false (N means Natural number, natural…

A:

Q: (9) Show that the given collection F is an open cover for S such that it does not contain a finite…

A: (9) (a) Given that S = 0, 2 and F = Un | n∈ℕ where Un = 1n, 2-1n. The objective is to show that the…

Q: (7) Let S = {1- (-)" | n € N}. Determine inf S and sup S. Justify. n

A: To find the supremum and infimum of the sequence S .

Q: (d) If each fn has fewer than M discontinuities (where M ∈ N is fixed), thenf has fewer than M…

A: To Show: If each fn has fewer than M discontinuities (where M ∈ N is fixed), then f has fewer than M…

Q: Q. No.3 (a) Show that v5 is an irrational number. (b) Let x and y are two irrational numbers. Then…

A:

Q: 6. Show that (x, y) = 1/2 (||x + y||² − ||t||² — ||ý||²) -

A: Given; &lt;x,y&gt;=(llx+yll2-llxll2-llyll2)/2.

Q: Let (t)= a. b. 2 53 (t² + 1 (t²-t 1|SL|SII e - e if 0 ≤ t < 2 if 2 st C ++ 211212 C. c. +-e-²5+/ d.…

A:

Q: If |Z| ≤ 1 and w) ≤ 1, then show that |zw|2s (1z|-|w|)² + (arg z - arg w)²2.

A:

Q: onal the (i) Int(A) (ii) (A) and A (iii) Whether or not A is open or closed.

A:

Q: 14. Let f(n) = n² + 20 and g(n) = n³. Show that f(n) is O(g(n)) by finding a positive integer N for…

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Q: 1.5 Prove that e-2=| 0.

A:

Q: 1. Let k and n be non-negative integers. (a) What does the function pr(n) count? (b) Show that the…

A: k and n are :Two non-negative integers.