1 To be Graded
Problem 1. Prove directly, by verifying the definition, that each of the following sequences
is a Cauchy sequence in the metric space (X, ρ) with X = R and ρ(x, y) = |x − y|:
(a) The sequence {xn} with xn =
1
√
n
;
(b) The sequence {xn} with xn =
cos n
2n
.
Problem 2. Let (X, σ) be a metric space and suppose that {xn} and {yn} are two Cauchy
sequences in X. Prove that the sequence of real numbers {sn}, defined as sn = σ(xn, yn),
converges in the usual Euclidean metric ρ(x, y) = |x − y|.
Problem 3. Let (X, σ) be a metric space and {xn} a Cauchy sequence in X. Let {yn} be
another sequence in X such that σ(xn, yn) → 0 (in the usual Euclidean metric ρ(x, y) =
|x − y|). Prove that
(a) {yn} is a Cauchy sequence;
(b) yn → y ∈ X iff xn → y ∈ X for the same y.
Problem 4. Let X be a non-empty set and ρ the discrete metric on X, meaning that:
∀x, y ∈ X, we have
ρ(x, y) = ?
0, if x = y
1, if x 6= y
Show that (X, ρ) is a complete metric space.
2 Reading Assignments
• Review Lecture Notes # 8 and # 9;
• Review Sections 2.2, 2.3 and 3.3 of the textbook;
