Math205a Fall Term 2002

Midterm Examination

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You have 50 minutes to solve the problems. Good luck!

1. Prove that ℝ \ ℚ is dense in ℝ. 2 2. Let x, y ∈ ℝ^{ℕ} be two sequences. Show that

lim sup_{k→∞}(x_{k} + y_{k}) ≤ lim sup_{k→∞} x_{k} + lim sup_{k→∞} y_{k} .

3. Let A ⊂ ℝ be compact and show that necessarily

inf A , sup A ∈ A .

Give an example of a non compact set with the same property.

4. Let a, b ∈ ℝ with a < b and a continuous function f : (a, b) → ℝ be

given with

f((a, b)) = {y_{1} , y_{2}} .

Prove that y_{1} = y_{2}.

5. Let x ∈ ℝ^{ℕ} be a convergent sequence with limit x_{∞} ∈ ℝ. Show that the sequence y ∈ ℝ^{ℕ} defined through

y_{n} = \frac{1}{n} ∑_{k=1}^{n} x_{k} , n ∈ ℕ ,

also converges to x_{∞}.