Math205c Spring Term 2003

Assignment 23

1. Let (x_{n})_{n∈ℕ} ∈ ℝ^{ℕ} and define X = {x_{n} | n ∈ ℕ}. Show by example that the limit points of X are accumulation points of (x_{n})_{n∈ℕ} but not vice-versa.

2. Let (M, d) be a metric space. Show that A ∪ B is compact whenever A, B ⊂ M are.

3. Let f ∈ C(ℝ, ℝ). Is it true that

f(lim sup_{n→∞} x_{n}) = lim sup_{n→∞} f(x_{n}) ?

4. Let f : D_{f} → ℝ be differentiable at x_{0} ∈ D_{f} ⊂^{o} ℝ and show that

lim_{h→0} \frac{f(x + h) − f(x − h)}{2h} = f′(x_{0}) .

The Homework is due Friday, May 23.