This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: MATH 242 Assignment 2 Solutions 1. Proof by induction on n . In case n = 1 the statement reduces to 1 + a 1 ≤ 1 + a 1 which is clearly true. So the induction starts. Now suppose that the statement is true for n we will establish it for n + 1 . We are assuming that 1 + n X j =1 a j ≤ n Y j =1 (1 + a j ) . Then n +1 Y j =1 (1 + a j ) = (1 + a n +1 ) n Y j =1 (1 + a j ) , ≥ (1 + a n +1 ) 1 + n X j =1 a j , since (1 + a n +1 ) ≥ = 1 + n X j =1 a j + a n +1 + a n +1 n X j =1 a j , ≥ 1 + n +1 X j =1 a j , since a n +1 ≥ and ∑ n j =1 a j ≥ . This completes the induction step. 2. This is a straightforward induction. For n = 1 the assertion reads 1 2 2 ≤ 1 4 , so the induction starts. For the induction step, we assume that the assertion is true for n and establish it for n + 1 . We find that 1 2 · 3 4 · 5 6 · . . . · 2 n 1 2 n · 2 n + 1 2 n + 2 2 = 1 2 · 3 4 · 5 6 · . . . · 2 n 1 2 n 2 · 2 n + 1 2 n + 2 2 ≤ 1 3 n + 1 · 2 n + 1 2 n + 2 2 , so it will be enough to establish 1 3 n + 1 · 2 n + 1 2 n + 2 2 ≤ 1 3 n + 4 (1) or equivalently (3 n + 4)(2 n + 1) 2 ≤ (3 n + 1)(2 n + 2) 2 . We find that (3 n + 4)(2 n + 1) 2 = 12 n 3 + 28 n 2 + 19 n + 4 and that (3 n + 1)(2 n + 2) 2 = 12 n 3 + 28 n 2 + 20 n + 4 so (1) follows since ≤ n . This completes the induction step. 3. (i) Clearly for any fixed j ∈ N we have inf ( j,k ) ∈ N × N a j,k ≤ inf k ∈ N a j,k since the inf on the left ranges over a larger set than the one on the right. Since the inf on the left is a lower bound for the gadgets on the right as j varies, we have inf ( j,k ) ∈ N × N a j,k ≤ inf j ∈ N inf k ∈ N a j,k (2) because the righthand side of (2) is the corresponding greatest lower bound....
View
Full
Document
This note was uploaded on 11/05/2009 for the course MATH MATH 242 taught by Professor Drury during the Fall '09 term at McGill.
 Fall '09
 Drury
 Math

Click to edit the document details