4130sols4

4130sols4 - 4130 HOMEWORK 4 Due Tuesday March 2 (1) Let N N...

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: 4130 HOMEWORK 4 Due Tuesday March 2 (1) Let N N denote the set of all sequences of natural numbers. That is, N N = { ( a 1 ,a 2 ,a 3 ,... ) : a i ∈ N } . Show that | N N | = |P ( N ) | . We use the Schr¨ oder-Bernstein Theorem. First, there is an injection from P ( N ) to N N , because we may regard a subset of N as a sequence of zeroes and ones, or equivalently as a sequence of 1’s and 2’s, and this gives the desired injection. The hard part is showing that there is an injection N N → P ( N ). To see this, note that a sequence of natural numbers is the same thing as a function N → N . But by definition, a function is a special kind of subset of N × N . In this way, we get an injection N N → P ( N × N ). But it was shown in class that N × N and N have the same cardinality, and hence so do their power sets. In this way, we get the desired injection N N → P ( N ). Working through the above proof, we can explicitly write down an injection if we like. An example is: ( a 1 ,a 2 ,... ) 7→ { 2 · 3 a 1 , 2 2 · 3 a 2 ,... } ⊂ N . (2) Let { x n } be a Cauchy sequence of rational numbers. Regarding { x n } as a sequence of real numbers, show that { x n } converges to the real number x defined as the equivalence class of the sequence { x n } . The hardest part of this is working out what to prove in the first place. Let ε > 0 be a real number. Choose A ∈ N with 3 / 2 A < ε . Since { x n } is a Cauchy sequence of rational numbers, there exists N ∈ N such that if m,n > N then | x n- x m | < 1 /A . In other words, if m,n > N then- 1 /A < x m- x n < 1 /A ....
View Full Document

This note was uploaded on 09/22/2010 for the course MATH 413 at Cornell.

Page1 / 5

4130sols4 - 4130 HOMEWORK 4 Due Tuesday March 2 (1) Let N N...

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

View Full Document
Ask a homework question - tutors are online