MATH 327A/574A: Introductory Real Analysis I Midterm Exam

12:30-1:20 pm., February 3, 2021

实分析期中代考 Use the definition of convergence (that is, the ε-N language)to  show  the  following.  Suppose the sequence {an} converges to a.

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This exam is closed book. However you can use one 8” × 11”-size sheet of notes, two-sided if you wish.  实分析期中代考

Show your work and justify your answers, i.e., show all reasoning. There are three  problems in  this exam.

You may use any results directly without a proof that the instructor has covered in the class or any theorems, propositions and lemmas in the main textbook by Fitzpatrick from Sections 1.1 up to Proposition 2.30 in Section 2.4 as well as those Propositions in Appendix A. You may also use the following property directly without a proof:   实分析期中代考

for two positive numbers x and y, x < y if and only if xⁿ < yⁿ for some n ∈ N. Do the problems that you can do first.

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Good Luck!

1.(10 points) Show that √5 is not a rational number.  实分析期中代考

2.(10 points) Use the definition of convergence (that is, the ε–N language)to  show  the  following.  Suppose the sequence {a_n} converges to a. Define the sequence {b_n} by

for every index n ∈ N.

Prove that the sequence {b_n} converges to a.  实分析期中代考

3 (10 points) Suppose that the sequence {a_n} is monotone. Prove that {a_n} converges if and only if {a2_n } converges.