Real Analysis I Homework # 05

1 To be Graded
Problem 1. Prove directly, using the definition of convergence, that each of the following
sequences converges in the metric space (X, ρ) with X = R and ρ(x, y) = |x − y|:
(a) The sequence {xn} with xn = 1 +
10
√
n
;
(b) The sequence {xn} with xn = 3 + 2−n
;
(c) The sequence {xn} with xn =
2n + 3
n + 1
.
Problem 2. Let (X, ρ) be a discrete metric space, and {xn} a sequence in X. Prove that
xn → x if and only if there exists a N ∈ N such that xn = x ∀n ≥ N.
Problem 3. Let ρ and σ be two uniformly equivalent metrics defined on X and {xn} be a
sequence in X. Show that xn → x in metric ρ iff xn → x in metric σ.
2 Reading Assignments
• Review Lecture Notes # 7 and # 8;
• Review Sections 2.2, 3.1 and 3.2 of the textbook;