Prove or Disprove 23. If x <y, then-y -x. Remark 39. What is the analogous problem...
Prove, or give a counter example to disprove the following statements. a) b) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
·J (I) < 0 for all such y. (Hint: let g(x)--f(x) and use part (a)) 3. In this problem, we prove the Intermedinte Value Theorem. Let Intermediate Value Theorem. Let f : [a → R be continuous, and suppose f(a) < 0 and f(b) >0. Define S = {t E [a, b] : f(z) < 0 for allェE [a,t)) (a) Prove that s is nonempty and bounded above. Deduce that c= sup S exists, and that astst (b) Use Problem...
Prove X, Y, Z, JER. XKY Prove Z <# X AND Y<j, if and only if (x,y) [z,j]
3. (a) Given n e N, prove that sup{.22 : 0<x<1} = 1 and inf{.22n: 0<x<1} = 0. (b) Find the supremum of the set S = {Sn: ,ne N}. Give a proof.
Problem 10(20). Let x and y be vectors in R". Prove that |x"y| < ||x|||y- No work, no credit, messy work, no credit, missed steps, no credit disorganized work, no credit.
Prove that A = B for: A = {(x,y) e Rº : +y/<1} B = {(z,y) € RP: (71+ y)² < 1}
1. Let a, b,cE Z be positive integers. Prove or disprove each of the following (a) If b | c, then gcd(a, b) gcd(a, c). (b) If b c, then ged(a., b) < gcd(a, c)
1. Let x, a € R. Prove that if a <a, then -a < x <a.
7. Is F(x, y) =< 2x, 6y? > a conservative vector field?
Prove or Disprove that: If a > 0 and b are two rational numbers, then a' is a rational number.