Prove that for positive real numbers x and y, the following inequality holds:
direct proof:
square of positive real quantity is greater than or equal to 0
Prove that for positive real numbers x and y, the following inequality holds:
(a) Let a,..., a, and by,...,b be real numbers. Prove the Schwarz inequality. That is, (b) When does equality hold in the Schwarz inequality?
Prove that for any two real numbers x and y, |x + y| ≤ |x| + |y|. Hint: Use the previously proven facts that for any real number a, |a|≥ a and |a|≥−a. You should need only two cases.
#2. Let n E N and X1,X2, ,yn, and zi,22, An be real numbers. ,An, yī,Y2, #a) Prove the identity #b) Use the identity in #a) to prove (the Cauchy-Schwartz inequality) that #1) Extend the result in #b) to prove that #d) Use the inequality in #b) to prove the inequality which is the triangle inequality
#2. Let n E N and X1,X2, ,yn, and zi,22, An be real numbers. ,An, yī,Y2, #a) Prove the identity #b) Use the identity...
use proof by induction
Day 1. Consider the inequality n 10000n. Assume the goal is to prove that inequality is true for all positive integers n. A common mistake is to think that checking the inequality for numerous cases is enough to prove that statement is true in every case. First, verify that the inequality holds for n-1,2,-.- ,10. Next, analyze the inequality; is there a positive integer n such that the inequality n 10000n is not true!
Day 1....
9. Let x,y > 0 be real numbers and q, r E Q. Prove the following: (а) 29 > 0. 2"а" and (29)" (b) x7+r (с) г а — 1/29. 0, then x> y if and only if r4 > y (d) If q (e) For 1, r4 > x" if and only if q > r. For x < 1, x4 > x* if and only if q < r.
Prove that x^2+xy+y^2≥0 for all real numbers, x and y. Find the values that result in equality.
#2. Let n E N and x1,x2,.., Xn, yı,y2,..,Ja, and zł,Zy, #a) Prove the identity An be real numbers #b) Use the identity in #a) to prove (the Cauchy-Schwartz inequality) that #1) Extend the result in #b) to prove that 4 #d) Use the inequality in #b) to prove the inequality which is the triangle inequality
#2. Let n E N and x1,x2,.., Xn, yı,y2,..,Ja, and zł,Zy, #a) Prove the identity An be real numbers #b) Use the identity in...
Prove that there is a unique ordered pair (x, y) where x and y are real numbers such that y=x’ and y=2x-1. (Be sure to prove both “existence" and "uniqueness.") (25 pts)
A random variable X obeys the exponential distribution law xp-x),(20). m! Prove that the following inequality holds true:
PROVE OR DISPROVE
1. If {an} is a non-increasing sequence of positive real numbers such that Σ" an converges, then lim na 0