## Question

###### THEORETICAL PROBLEMS= (1-4) Answer each of the following statements use any proof technique that You would like to use) For every nonnegative real number and natural number n. (1 +x)" > 1+nx_ Use the well-ordering principle to show that there is no natural number x such that 0 Hint: Do not use induction O the well-ordering proof technique, use the well- ordering principle directly: Prove that if 23 + 2x2 0. then 2x + 5 <11. Prove that the limit of the following sequence is lilln-+o &q

THEORETICAL PROBLEMS= (1-4) Answer each of the following statements use any proof technique that You would like to use) For every nonnegative real number and natural number n. (1 +x)" > 1+nx_ Use the well-ordering principle to show that there is no natural number x such that 0 Hint: Do not use induction O the well-ordering proof technique, use the well- ordering principle directly: Prove that if 23 + 2x2 0. then 2x + 5 <11. Prove that the limit of the following sequence is lilln-+o "+3 n3+1 = 0 Let a. b, and c be integers. Prove that if c2 + 62 22 , then at least one of & and b is even Prove that for all natural numbers n > 1. 2" < 2n+1 2"-1 _ -1. Prove that 10n+1 + 3.4"-1 + 5 is divisible by 9 for all natural numbers n 2 1. Let a, b, and be integers Prove that if a k bc; then *b. Prove that for all natural numbers n, 1 + {+}+f+3+ +321+3 Hint: Unlike previous formulae for sums you ve had to prove; the number of additional terms at each step as n increases is more than Let A, B, and â‚¬ be sets. Prove that (4 B) U (A C) = A iff AnBnc = 0. Let a, b, and c be odd integers and a real solution to az2 _ br +e= 0. Prove that ~is irrational Let a, b. and be positive integers. Prove that if ab 2 c then 2 Vc o1 b 2 Vc.