1 To be Graded
Problem 1. Let x, y ∈ R be such that x 6= y. Prove that ∃ε 0 such that Bε(x)∩Bε(y) = ∅.
Problem 2. Let X = [0,∞) and take Ok = (−10, k), k ≥ 1. Prove that:
(i) O = {Ok}
∞
k=1 is a open cover of X;
(ii) O has no finite subcover, i.e. X is not compact.
Problem 3. Let (X, ρ) be a metric space with ρ the discrete metric. Prove that (X, ρ) is
compact if and only if X is a finite set.
Problem 4. Let (X, σ) be a metric space, and f(x) : Dom(f) = X 7→ R be a continuous
function (under the usual Euclidean metric ρ(x, y) = |x − y| on R). Prove that |f(x)| is a
continuous function on Dom(f).
Problem 5. Let (X, σ) be a metric space, f : X → R and g : X → R be both Lipschitz
on X (under the usual Euclidean metric ρ(x, y) = |x − y| on R). Prove that f + g is also
Lipschitz on X.
Problem 6. Let (X, σ) be a metric space, f : X → R and g : X → R be both uniformly
continous on X (under the usual Euclidean metric ρ(x, y) = |x − y| on R). Prove that f + g
is also uniformly continous on X.
2 Reading Assignments
• Review Lecture Notes # 10 and # 11;
• Review Sections 2.3, 4.1, 4.2 and 4.3 of the textbook;
