{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

4130prelim1

# 4130prelim1 - L(c(8 marks Let x 1 ∈ R and deﬁne a...

This preview shows page 1. Sign up to view the full content.

MATH 4130 HONORS INTRODUCTION TO ANALYSIS I. PRELIM 1. THURSDAY MARCH 11 2010 Please attempt all questions. You have 70 minutes. You may use any theorems from the lecture notes, but please clearly state any theorems which you use. (1) (9 marks) Let X = (0 , 1) [3 , 4] R . State whether the following statements about X are true or false and give a brief reason in each case. (a) sup( X ) = 4. (b) X can be written as a union of open sets. (c) | X | = | R | . (2) (19 marks) Let { x n } be a sequence of real numbers. (a) (3 marks) State what it means for { x n } to converge to the limit L R . (b) (8 marks) Let k N and define a sequence { y n } by y n = x n + k , n 1. Suppose { x n } converges to L . Show that { y n } also converges to
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: L . (c) (8 marks) Let x 1 ∈ R and deﬁne a sequence of real numbers { x n } by x n +1 = x 2 n + x n + 1 , n ≥ 1 . Show that the sequence { x n } does not converge. (3) (22 marks) Let A ⊂ R . (a) (3 marks) Explain what it means to say that x ∈ R is a cluster point (a.k.a. limit-point; accumulation point) of A . (b) (3 marks) Explain what it means to say that the set A is bounded . (c) Now let S = { x ∈ R : x is a cluster point of A } . (i) (8 marks) Show that S is a closed set. (ii) (8 marks) Suppose A is bounded. Show that S is a compact set. [END OF PAPER]...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern