Advanced Calculus Midterm Exam
November 4, 2019 1. Let z and w be complex numbers.(a) (4 points) Prove that |Re z| ≤ |z|.
(b) (4 points) Prove that |z + w| ≤ |z| + |w|.
2. Suppose S is an ordered set with the least-upper-bound property and E is a nonempty subset of S.
(a) (4 points) If α = sup E exists, prove that it is unique.
(b) (4 points) If E ⊂ F ⊂ S and F is bounded above, prove that E is bounded above and sup E ≤ sup F.
3. (a) (10 points) If x, y ∈ R and x > 0, prove that there is a positive integer n such that nx> y.
(b) (10 points) If x, y ∈ R and x < y, prove that there exists a p = m
n ∈ Q, n ∈ N, such that x< p < y ⇐⇒ x < m
n < y ⇐⇒ nx < m < ny.
4. (8 points) For x , y ∈ R, define the function d : R × R → [0, ∞) by d(x, y) = |x − y|
1 + |x − y|. Prove that d is a metric.
5. Let (X , d) be a metric space and let E ⊂ X .
(a) (8 points) If p ∈ E, prove that p is either an interior point or a boundary point of E.
(b) (8 points) If p is a limit point of E ⊆ X = (X , d), prove that every neighborhood of p contains infinitely many points of E.
6. Let X = (X , d) be a metric space.
(a) (8 points) If {Uα} is a collection of open sets in X, prove that [
α
Uα is open in X .
(b) (8 points) If K is a compact subset of X , prove that K is closed.
(c) (8 points) If K is a compact subset of X , prove that every infinite subset S of K has a limit point in K, i.e if S is a subset of K containing infinitely many elements, prove that S0∩ K 6= /0.
7. (a) (8 points) Let E be a nonempty proper subset of a metric space X , i.e. E ⊂ X , E 6= /0 and E 6= X . If E is a both open and closed in X , prove that X is disconnected.
(b) (8 points) Let E be a subset of R. If E is connected, prove that E has the “interval property”: if x, y ∈ E and x < z < y, then z ∈ E.
