实分析代写 Real Analysis代写 Class Test代写

MT3502 Real Analysis – Class Test

实分析代写 It would be helpful if you would start each question on a separate sheet of paper. Section A – Multiple choice Q1. Are the following

Answer all questions.

It would be helpful if you would start each question on a separate sheet of paper.

Section A – Multiple choice 实分析代写

Q1. Are the following statements True or False? Each part carries 1 mark with no penalities for incorrect answers

Please write your answers in a column (do not include any justification), e.g.

(a) True

(b) False

.

.

.

(a) sup_x∈[0,1](f(x) + g(x)) = sup_x∈[0,1] f(x) + sup_x∈[0,1] g(x), for all bounded functions f, g : [0, 1] → R.

(b) If (x_n) is a Cauchy sequence with x_n ∈ (0, 1) for all n then (x_n) converges to some x ∈ (0, 1).

(d) If f, g : R → R are both uniformly continuous on R then fg : R → R is uniformly continuous on R.

(e) If f : R → R satisfies |f(x) − f(y)| ≤ 8|x − y| for all x, y ∈ R then f is uniformly continuous on R.

(j) Let f, g : [0, 1] → R with 0 ≤ g(x) ≤ f(x) for all x ∈ [0, 1]. If f is integrable then g is integrable.

Section B – Written answers 实分析代写

Each of the three questions in this section carries 5 marks. You should carefully justify your answers.

Q2. Let X be the set of all strictly increasing sequences of natural numbers, that is

X = {(a₁, a₂, a₃, . . .) : a_n ∈ N and a_n < a_n+1 for all n ∈ N} .

Show that X is uncountable.

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