In this post, I will discuss the following definitions we covered in Thursday’s (9/7) class:

Chapter 1: Theorem 1.5.6

Chapter 2: Sequence, Convergence of a Sequence, Theorem 2.2.7 (Uniqueness of Limits) and Bounded

 

Theorem 1.5.6 (i) Q ∈ N (Q is countable) (ii) R is uncountable

(Proof Template)

(i) Set A1 = {0} and for each n ≥ 2, let An be the set given by An = {± p/q: where where p, q ∈ N are in lowest terms with p + q = n}

The first few of these sets look like A1 = {0}, A2 ={1/1, -1/1}, A3= {1/2, -1/2, 2/1, -2/1} and A4 ={1/3, -1/3, 3/1, -3/1}

We can observe that each An is finite and every rational number appears in exactly one of these sets. Our 1–1 correspondence with N is then achieved by consecutively listing the elements in each An.

(ii) Proof by contradiction. Assume that there does exist a 1–1, onto function f : N → R. Let x1 = f(1), x2 = f(2)… R = {x1, x2, x3, x4, . . .} We can use the Nested Interval Property (Theorem 1.4.1) to produce a real number that is not there.

 

Sequence: A sequence is a function whose domain is N

 

Convergence of a Sequence: A sequence (a_n) converges to a real number a if, for every positive number ε, there exists an N ∈ N such that whenever n ≥ N it follows that |a_n − a| < ε.

(i) lim a_n = a or lim_(n→∞) a_n = a

(ii) (a_n) converges in R

Notation: ∀ ε > 0, ∃n_0 ∈ N such that |a_n − a| < ε for all n ≥ n_0.

 

Convergence of a Sequence (Topological Version): ∀ ε > 0, ∃ n_0 ∈ N such that a_n ∈ V_ε(a) for all n ≥ n_0.

 

Theorem 2.2.7 (Uniqueness of Limits): The limit of a sequence, when it exists, must be unique.

Notation: (a_n) → a and (a_n) → b, then a = b.

(Proof Template)

Recalls Theorem 1.2.6 ∀ a, b ∈ R, a = b ↔  ∀ ε > 0, |a − b| < ε.

PF: let ε > 0. Since (a_n) → a, ∃ N1 ∈ N such that |a_n − a| < ε/2 for n ≥ N1. Since (a_n) → b, ∃ N2 ∈ N such that |a_n − b| < ε/2 for n ≥ N2. Pick n_0 = max{N1, N2} and consider n ≥ n_0. |a − b| = |a – a_n + a_n − b| ≤ |a – a_n| + |a_n – b| < ε/2. Since n ≥ n_0 ≥ N1, |a – a_n| < ε/2. Since n ≥ n_0 ≥ N, |a_n – b| < ε/2. Thus, since ε > 0 is arbitrary, we conclude a = b.

 

Bounded: (a_n) is bounded if ∃ M > 0 so that |a_n| ≤ M, ∀ n ∈ N.