FINAL FOR ADVANCED CALCULUS
Instructor: Shu-Yen Pan
No credit will be given for an answer without reasoning.
1.
(i) Find a sequence an with lim sup an= 5 and lim inf an = −3.
(ii) Give an example of a contraction map Φ : R2→ R2with fixed point (1, 1).
2. Let A, B be two non-empty subsets of R.
(i) Is sup(A ∪ B) ≥ sup{sup(A), sup(B)}? Prove it or give a counterexample.
(ii) Is sup(A ∪ B) = sup{sup(A), sup(B)}? Prove it or give a counterexample.
3. Suppose that f : [0, 1] → [0, 1] is continuous and onto. Show that there exists an x0∈ [0, 1] such that f (x0) = x0.
4. Suppose thatP∞
k=1ak= α (C, 1) and P∞
k=1bk= β (C, 1). Show thatP∞
k=1(ak+ bk) = α + β (C, 1).
5. Let f : A → N be continuous and let K ⊆ A be a compact set. Prove that f is uniformly continuous on K.
6. A subset A of R2 is called convex if x, y ∈ A implies tx + (1 − t)y ∈ A for all t ∈ [0, 1]. Show that a convex subset of R2is connected.
7. Show that the following set
A = { f ∈ C([0, 1], R) | 0 ≤ Z 1
0
f (x) dx ≤ 3 }
is closed in C([0, 1], R).
8.
(i) Is the set A in problem 7 bounded? Why or Why not?
(ii) Let f (x) = x2+ 1 and g(x) = x. Compute d(f, g) in C([0, 1], R).
9. Suppose that f is a differentiable function and α is a constant. Use definition to show that cf is also differentiable and D(αf ) = αDf .
10. Suppose that ak ≥ 0 for all k andP∞
k=0ak = α. Prove thatP∞
k=0akxk converges for |x| < 1 and limx→1−
P∞
k=0akxk = α by using the Weierstrass M -test.
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