M367K First Midterm Exam, February 7, 2008
1. Prove that every positive integer can be written as a sum of distinct
powers of 2. (For instance, 13 = 1 + 4 + 8.)
2. A real number is algebraic if it is a root of a polynomial with integer
coecients. For inst
M367K Second Midterm Exam Solutions, April 8, 2008
1. Comparing topologies. On RZ+ , consider the metric d(x, y ) = n 2n d(xn , yn ),
where d(xn , yn ) = min(|xn yn |, 1) is the standard bounded metric on R. Let
T be the metric topology generated by d and
M367K Second Midterm Exam, April 8, 2008
1. Comparing topologies. On RZ+ , consider the metric d(x, y ) = n 2n d(xn , yn ),
where d(xn , yn ) = max(|xn yn |, 1) is the standard bounded metric on R. Let
T be the metric topology generated by d and let T be