MATH 410 Study Guide - Midterm Guide: Archimedean Property

• School: University of Maryland
• Department: Mathematics
• Course: MATH 410
• Professor: All

Document Summary

Each problem below is worth 15 of these points. The take home part is worth 25 points: for each of the results (1)-(3), write down the letter corresponding to the main ingredient in its proof. (no explanation required. ) Ingredients: (a) archimedean property of r (b) bolzano-weierstrass theorem (c) completeness axiom for r. Let a = sup a and let b = sup b. Prove that a + b is the least upper bound of the set a + b: de ne the sequence {sn} by the rule sn = a1 +a2 + +an, where an = 1/(n2n). Prove that the sequence {sn} converges: suppose f : r r is continuous and f (x0) = 0.