实分析数学作业代写 Math 441/541代写

Math 441/541 Real Analysis

Homework # 5

Homework # 6  实分析数学作业代写

1. Let K be a non-empty compact subset of R. Prove that sup K and inf K both exist and both are elements of K.
2. Let K be a non-empty compact subset of R and let y ∈ R with y∉K. Prove that there exists elements a and b in K such that

|a − y| = inf{|x − y| : x ∈ K} and |b − y| = sup{|x − y| : x ∈ K}.

3. Prove that the union of a finite number of compact subsets of R is compact. Does a countable union of compact subsets of R have to be compact? Explain your answer.
4. Let A and B be two non-empty compact subsets of R. Define

d(A, B) = inf{|a − b| : a ∈ A, b ∈ B}.

Prove that A ∩ B ≠ ∅ if and only if d(A, B) = 0.

Homework # 7  实分析数学作业代写

1. Let f : [a, b] → R be a continuous function on [a, b] with f([a, b]) ⊆ [a, b]. Prove that there exists a point x ∈ [a, b] such that f(x) = x.
2. Suppose that K is a compact subset of R and f : K → R is continuous on K. Prove that {x ∈ K : 0 ≤ f(x) ≤ 1} is a compact set.
3. Prove that f : [0, ∞) → R defined by f(x) = √x is uniformly continuous on [0, ∞).

更多代写：cs北美网课代做  托福作弊被抓  英国统计代上网课价格   北美MKT代写  北美Chemistry化学quiz代考  软件原理家庭作业代写

合作平台：essay代写 论文代写 写手招聘 英国留学生代写