Homework 3. Due on October 9

1. Find the interior points, the closure of the following sets; justify your answer:

(a) E := \{\frac{1}{2^{n}} |n ∈ ℕ\}

(b) E := (ℚ ∩ (1, 3)) ∪ {4} ∪ [5, 6)

2. Show that the intervals (a, ∞) and (−∞, a) are open sets, and that the intervals [b, ∞) and (−∞, b] are closed sets.

3. Exhibit an open cover of (1, 2] which does not have a finite subcover. Justify your example.

4. Let E ≢ ∅ be a closed and bounded set in ℝ and let c ∈ ℝ. Prove that there exists a point a ∈ E such that |c − a| = inf{|c − x|x ∈ E}.

5. Use the definition of the limit of a sequence to find the limit of the following:

(a) lim ( \frac{1}{\sqrt{n+3}} ),

(b) lim ( \frac{n^{2}+5}{3n^{2}−2} )

(c) lim ( \frac{1}{3^{n}} )

6. Prove that lim(x_{n}) = x then lim(|x_{n}|) = |x|, but the converse statement does not hold.

7. For x_{n} given by the following formulas, establish either the convergence or the divergence of the sequence X = (x_{n}).

(a) x_{n} := \frac{n}{n+2}. (b) x_{n} := \frac{(−1)^{n}n}{n+1}. (c) x_{n} := \frac{(−1)^{n}}{n+3} .

8. Probelm 9 (a), (b) on page 43 of Rudin.

9. Problem 2, on page 78 of Rudin.

10. Problem 3, on page 78 of Rudin. What do you think is the limit of the sequence?