Western University

MATH 2122

IMPORTANT: When giving definitions you may assume that all other definitions in the course are
known. When giving a proof you may use any definition or theorem in the course. No aids are
permitted. Each part of each question is worth 5 marks. Answer all questions.
ONLY FINAL ANSWERS should be written in the space provided below each question. For draft
solutions and other preliminary work use the backs of the test pages, or scrap paper provided.
1. a) Let (an) be a sequence of real numbers. Define the phrase,\(an) is bounded".
(an) is bounded above and below (OR) the underlying set of (an) is bounded (OR) there exists an
M 2 R such that janj ≤ M for all n 2 N+ (OR) there exist m; M 2 R such that m ≤ an ≤ M for all
n 2 N+ (OR) ...
1. b) Let (an) be a sequence of real numbers. Set bn = a2 n=(1 + a2 n) for n 2 N+. Prove that (bn) has
a convergent subsequence.
Since 0 ≤ a2 n ≤ a2 n + 1 for each n, 0 ≤ bn ≤ 1 for each n. Thus, (bn) is bounded. By the BolzanoWeierstrass Theorem, (bn) has a convergent subsequence.
2. a) Let (an) be a sequence of real numbers. Define the phrase, \(an) is a Cauchy sequence".
For all " > 0 there exists an N 2 N+ such that if m; n ≥ N then jam - anj < ".
2. b) Let an 2 R for all n 2 N+. Suppose jam - anj ≤ n 1 + m 1 for all m; n 2 N+. Prove that (an)
converges.
Fix " > 0 and let N be an integer greater than 2=". If m; n ≥ N then jam - anj ≤ n 1 + m 1 ≤ N 2 < ".
Thus, (an) is Cauchy and therefore convergent