MATH 5070-001: APPLIED ANALYSIS
Fall 1998, University of Colorado at Denver
Instructor: Prof. Andrew V. Knyazev

Test 1

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

1. Is the sum of the series

   

   rational, or irrational?

2. Consider a set of all sequences such that every element of the sequence is either 0, or 2. Determine if the set of sequences is finite, countable, or uncountable.
3. Consider the metric space equipped with the standard euclidean metric. In this metric space, is the set

   

   closed? Open?

4. Is the set of all prime numbers compact in the metric space R?