The greatest lower bound for S is denoted by inf S

1. hw 5

Recall: Let S be a subset of R. We say that S is bounded below if there exists a ∈ R such that x ≥ a for any x ∈ S. In this case, a is called a lower bound for S. Assume that S is bounded below.

We say that a lower bound L for S is the greatest lower bound for S if a is a lower bound for S, then a ≤ L. The greatest lower bound for S is denoted by inf S.

(1) A subset S of Rⁿ is finite if number of elements of S are finite. Show that any finite subset of Rⁿ is closed.

(2) Let A be a finite subset of Rⁿ and

B = {x ∈ Rⁿ: d(x, y) ≤ 1 for some y ∈ A}.

Show that B is closed.

(3) Let A be a subset of Rⁿ and A⁰ be its derived set (the set of all accumulation points) of A.

Prove that A⁰ is closed. Is (A⁰)⁰= A⁰ for all A?

(4) Let S = {(r, s) ∈ R²: r, s ∈ Q}.

(a) Find S.

(b) Find S^c. Here S^c= R²\ S.

(c) Find ∂S.

(5) We have two equivalent definitions of convergence of sequences in R^p.

Definition 1.1. (Defn I) A sequence (an) in R^p is convergent to a point a ∈ R^p if for any

 > 0, there exists a natural number N (this natural number depends on the choice of ) so that

a_n∈ B(a, ) whenever n ≥ N.

Definition 1.2. (Defn II) A sequence (an) in R^pis convergent to a point a ∈ R^pif for any neighborhood U of a, there exists a natural number NU of a such that

a_n∈ U whenever n ≥ N.

We have proved in class that Defn I is equivalent to Defn II. Students may not be familiar with Defn II. In this exercise, we will let you experience how NU depends on the choice of U.

Let (a_n) be the sequence defined by an=

 sin1

n,1 n



, n ∈ N.

(a) Use Defn I to show that (an) is convergent to 0 = (0, 0).

(b) Let A = {(x, y) : x²+ 4y²≤ 10⁻⁵}. Show that A is a neighborhood of 0, i.e. find  > 0 so that B(0, ) ⊂ A. (A is a neighborhood of 0 if and only if 0 is an interior point of A.) Use this  to find a natural number N so that an∈ A for all n ≥ N. This number will denoted by NA.

(c) Let B =



(x, y) : − 1

10⁶ ≤ x, y ≤ 1 10⁶



. Show that B is a neighborhood of 0 i.e. find δ > 0 so that B(0, δ) ⊂ B. Use δ to find a natural number M so that aⁿ ∈ B for all n ≥ M. This number will denoted by M = NB.

(6) Prove that the following sequences have a convergent subsequence in the given spaces.

(a) Let (a_n) be the sequence of real numbers in R defined by a_n= sin n, n ∈ N.

(b) Let (bn) be the sequence of real numbers in R defined by bn= (−1)ⁿsin eⁿ, n ∈ N.

1

2

(c) Let (an) be the sequence in R²defined by

an = (cos n, sin n), n ∈ N.

(d) Let (b_n) be the sequence in R³ defined by b_n=



cos n, e⁻ⁿcos n², cos1 n



, n ∈ N.