MATH 242 Midterm: MATH242 Fall 2010 Exam

• School: McGill University
• Department: Mathematics and Statistics
• Course: MATH 242
• Professor: All

Document Summary

1i(6 marks) the archimedean property states that the set of nat- ural numbers n is unbounded in r. assuming this property, show that inf{1/n : n n} = 0. Conclude that if t > 0, there exists some n n with 1 n < t, and if y > 0, there exists a natural number n such that n 1 y < n. 1ii(6 marks) use 1i to show that for any two real numbers x and y with x < y, there exists a rational number r with x < r < y. 2i(3 marks) let f : a r be a function. De ne what it means for f to be continuous on a, and uniformly continuous on a. 2ii (4 marks) show that if f and g are uniformly continuous func- tions on a, and they are both bounded on a, then their product f g is uniformly continuous on a.