HW16solns

# Mathematical Thinking: Problem-Solving and Proofs (2nd Edition)

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 13.12 This is false. Let S = { , 1 } and set x n = 1 ,y n = 0. 13.23 Since sup A ≥ a and sup B ≥ b for every a ∈ A and b ∈ B , sup A +sup B ≥ a + b for every a + b ∈ C . Thus sup A +sup B is an upper bound for C . By Proposition 13.15, there are sequences h a n i in A and h b n i in B such that a n → sup A and b n → sup B . Then h a n + b n i is a sequence in C with a n + b n → sup A + sup B . By Proposition 13.15, sup A + sup B = sup C . 13.29 First we prove that h x n i converges using the Monotone Convergence Theorem. Clearly 0 ≥ x n . On the the other hand, x n = 1 + n 1 + 2 n = 1 n + 1 1 n + 2 ≤ 1 n + 1 2 ≤ 1 + 1 2 = 2 . Therefore h x n i is bounded. Further, for every n ∈ N we have x n +1- x n = 1 + ( n + 1) 1 + 2( n + 1)- 1 + n 1 + 2 n = 2 + n 3 + 2 n- 1 + n 1 + 2 n = (2 + n )(1 + 2 n )- (1 + n )(3 + 2 n ) (3 + 2 n )(1 + 2 n ) = (2 n 2 + 5 n + 2)- (2 n 2 + 5 n + 3) (3 + 2 n )(1 + 2 n ) =- 1 (3 + 2 n )(1 + 2 n ) < , so h x n i is monotone. By the Monotone Convergence Theorem,is monotone....
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

HW16solns - 13.12 This is false Let S = 1 and set x n = 1,y...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online