Math 205a Fall Term 2002

Assignment 2

1. Let (x_{n})_{n∈ℕ} ∈ CS(ℚ) and consider the polynomial function defined through

p(x) = ∑_{j=1}^{m} p_{j}x^{j} , x ∈ ℚ ,

with rational coefficients p_{j} ∈ ℚ , j = 1, …, m. Prove that the sequence (p(x_{n}))_{n∈ℕ} ∈ CS(ℚ) , too.

2. Describe the set CS(ℕ) and determine whether it is countable or not. [Hint: Prove first that the countable union of countable sets is countable.]

3. Prove that ℝ is uncountable and has the same cardinality as 2^{ℕ}.

4. Prove that

|x − y| ≥ ||x| − |y|| ∀x, y ∈ ℝ .

The Homework is due Friday, October 18