By Brezis H.

ISBN-10: 2225771987

ISBN-13: 9782225771989

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J21 < 13. Prove the Polynomial Property for convergent sequences by using an inductive argument based on the degree of the polynomial. 14. Define the sequence {sn} by 1 = -2·1 Sn 1 1 + -- + · · · + - - - 3·2 (n + l)(n) for every index n. Prove that lim n-+oo Sn = 1. 15. Let {an} be a sequence of real numbers. Suppose that for each positive number c there is an index N such that for all indices n ::=:: N. When this is so, the sequence {an} is said to converge to infinity, and we write lim an= n-+oo 00.

Use this property to show that the sequence converges to J2. J21 < 13. Prove the Polynomial Property for convergent sequences by using an inductive argument based on the degree of the polynomial. 14. Define the sequence {sn} by 1 = -2·1 Sn 1 1 + -- + · · · + - - - 3·2 (n + l)(n) for every index n. Prove that lim n-+oo Sn = 1. 15. Let {an} be a sequence of real numbers. Suppose that for each positive number c there is an index N such that for all indices n ::=:: N. When this is so, the sequence {an} is said to converge to infinity, and we write lim an= n-+oo 00.

Prove that if n and m are natural numbers such that n > m, then n - m is also a natural number. ) 9. Use Exercise 8 to prove that the sum, difference, and product of integers also are integers. 10. Use Exercise 9 to prove that the rational numbers satisfy the Field Axioms. 11. a. Prove that the sum of a rational number and an irrational number must be irrational. b. Prove that the product of two nonzero numbers, one rational and one irrational, is irrational. 12. 2 to show that there is no rational number whose square equals 2/9.

### Analyse fonctionnelle by Brezis H.

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