MATH 5070-001: APPLIED ANALYSIS
Fall 1996, University of Colorado Denver
Instructor: Prof. Andrew V. Knyazev

Test 1

Please, provide complete and detailed proofs of every statement, or give a reference to the book if you use a statement from the book.

1. Prove that the number is irrational.
2. Prove that the set of all complex numbers z on the unit circle, i.e. such that |z|=1, with rational real parts is countable.
3. We consider a metric space equipped with a new metric

   

   

   Prove that the metric is homeomorphic to the standard euclidean metric

   

4. Find the closure of a set A of all positive rational numbers smaller then 1, i.e.

   

   in the metric space of all rational numbers Q with the standard metric r(x, y) = | x-y |.

5. For extra credit.

   Let M=QxR be a metric space of ordered pairs z=(x y) with any rational x and any real y, equipped with the metric

   

   

   Consider a set A of all pairs of positive rational numbers smaller then 1, i.e.

   

   and find its closure in M.