4. Define the functions fn : 1-1, 1] → R given by TL Prove that fn → Irl uniformly on I-1, 1]. Note that the limit function Irl is a continuous function but not differentiable at r-0.
Q3 (Prove that P∞ k=1 1/kr < ∞ if r > 1) . Let f : (0,∞) → R be a twice differentiable function with f ''(x) ≥ 0 for all x ∈ (0,∞). (a) Show that f '(k) ≤ f(k + 1) − f(k) ≤ f '(k + 1) for all k ∈ N. (b) Use (a), show that Xn−1 k=1 f '(k) ≤ f(n) − f(1) ≤ Xn k=2 f '(k). (c) Let r > 1. By finding...
Question 15: Let n 〉 r 〉 1 . Prove that any n-vertex graph of minimum degree more than n -n/r contains Kr+1 without using Turán's Theorem. Question 15: Let n 〉 r 〉 1 . Prove that any n-vertex graph of minimum degree more than n -n/r contains Kr+1 without using Turán's Theorem.
Q2 (m) = n/(m + n). Prove that :N → R by define 2. For n (m) = n/(m + n). Prove that :N → R by define 2. For n
49.12. Let G be a graph with n 2 2 vertices. a. Prove that if G has at least ("21) +1 edges, then G is connected. b. Show that the result in (a) is best possible; that is, for each n 2 2, prove there is a graph with ("2) edges that is not connected. 49.12. Let G be a graph with n 2 2 vertices. a. Prove that if G has at least ("21) +1 edges, then G is...
Prove that Şi = n(n+1) for all integers n 2 1.
prove n^3 >=(n+1)^2 for all n>=2 Step by step answer would be appreciated. Prove that the following statement is true oo give counter example, h3> (htig for all nya
help please and thank you 5. Prove that --> 2(n+1 - 1) for all n e Zt. 6. Prove that n < 2" for all n e Z.
2. (15 points) Prove that for a positive integer n, the number gcd (n + 1, na — n + 1) is equal either to 1 or to 3.
1. Prove the following statement by mathematical induction. For all positive integers n. 2++ n+1) = 2. Prove the following statement by mathematical induction. For all nonnegative integers n, 3 divides 22n-1. 3. Prove the following statement by mathematical induction. For all integers n 27,3" <n!