Math 205a Fall Term 2002

Assignment 3

1. Let (k_{n})_{n∈ℕ} ∈ ℕ^{ℕ}. For any m ∈ ℕ define

x_{m} = k_{1} + \frac{1}{k_{2} + \frac{1}{k_{3} + \frac{1}{⋮ ; k_{m−1} + \frac{1}{k_{m}}}}} .

Prove that (x_{m})_{m∈ℕ} ∈ CS(ℚ) and that any x ∈ [0, ∞) is obtained as a limit of such a sequence.

2. Let (x_{n})_{n∈ℕ} ∈ ℝ^{ℕ} with x_{n} ≤ x_{n+1} ≤ c < ∞ ∀n ∈ ℕ and show that there exists x_{∞} ∈ ℝ such that lim_{n→∞} x_{n} = x_{∞}. [Hint: Show that it is a Cauchy sequence.]

3. Let (x_{n})_{n∈ℕ} , (y_{n})_{n∈ℕ} ∈ ℝ^{ℕ} converge to the common limit x_{∞}. Prove that any sequence (z_{n})_{n∈ℕ} ∈ ℝ^{ℕ} with

x_{n} ≤ z_{n} ≤ y_{n} ∀ n ≥ m

for some m ∈ ℕ also converges to the same limit x_{∞}.

4. Construct sequences x = (x_{n})_{n∈ℕ} ∈ ℝ^{ℕ} such that (i) LP (x) = ℤ. (ii) LP (x) = {y} for some y ∈ ℝ but x is not convergent. (iii) LP (x) = [0, 1] .

The Homework is due Friday, October 25