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8. Prove that n Ei i! = (n+1)! – 1 i=1 for all positive integers n.

8. Prove that n Ei i! = (n+1)! – 1 i=1 for all positive integers n.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter7: Real And Complex Numbers
Section7.1: The Field Of Real Numbers
Problem 26E: Prove that if and are real numbers such that , then there exist a rational number such that ....
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question8 induction
1. Prove by contradiction that 6n + 5 is odd for all integers n. 2. Prove that for all integers n, if 3n + 5 is even then n is odd. (Hint: prove the contra- positive) 3. Prove that |x + y| < ]x| + \y] for all real numbers x and Y. 4. Prove that there does not exist a smallest positive real number. (In other words, prove that there does not exist a positive real number x such that x < y for all positive real numbers y). 5. Recall that an irrational number is a real number which is not rational. Prove that if x is rational and y is irrational, then x+y is irrational. You may use the fact that the rational numbers are closed under addition - if a and b are rational numbers, then a + b is rational as well. 6. Prove that if a, b, and c are positive real numbers with ab = c, then a <Vc or b < Vc. 7. Prove that 1 i=1 for all positive integers n. 8. Prove that i · i! = (n+ 1)! – 1 i=1 for all positive integers n. 9. Prove that 2" < n! for all positive integers n such that n > 4.
Transcribed Image Text:1. Prove by contradiction that 6n + 5 is odd for all integers n. 2. Prove that for all integers n, if 3n + 5 is even then n is odd. (Hint: prove the contra- positive) 3. Prove that |x + y| < ]x| + \y] for all real numbers x and Y. 4. Prove that there does not exist a smallest positive real number. (In other words, prove that there does not exist a positive real number x such that x < y for all positive real numbers y). 5. Recall that an irrational number is a real number which is not rational. Prove that if x is rational and y is irrational, then x+y is irrational. You may use the fact that the rational numbers are closed under addition - if a and b are rational numbers, then a + b is rational as well. 6. Prove that if a, b, and c are positive real numbers with ab = c, then a <Vc or b < Vc. 7. Prove that 1 i=1 for all positive integers n. 8. Prove that i · i! = (n+ 1)! – 1 i=1 for all positive integers n. 9. Prove that 2" < n! for all positive integers n such that n > 4.
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