(1)

Let $\displaystyle B= \{\frac{(-1)^{n}n}{n+1}: n =1,2,3,...\}$

(a) Find the limit points of $\displaystyle B$.

(b) Is $\displaystyle B$ a closed set?

(c) Is $\displaystyle B$ an open set?

(d) Does $\displaystyle B$ contain any isolated points?

(e) Find $\displaystyle \overline{B}$

(For (e), use this definition. Given a set A $\displaystyle \subseteq R$, let L be the set of all limit points of A. Theclosureof A is defined to be $\displaystyle \overline{A}$= A $\displaystyle \cup$ L

(2) Decide wheather the following sets are open, closed, or neither. If a set is not open, find a point in the set for which there is no $\displaystyle \epsilon$-neighborhood contained in the set. If a set is not closed, find a limit point that is not contained in the set.

(a) $\displaystyle Q$-rational numbers

(b)$\displaystyle N$-natural numbers

(c) $\displaystyle \{x\in R: x>0\}$

(d) (0,1]=$\displaystyle \{x\in R: 0<x\underline{<}1\}$

(e) {1+1/4+1/9+...+1/$\displaystyle n^2: n \in N$}

(3) Let $\displaystyle a \in A.$ Prove that $\displaystyle a$ is an isolated point of A if and only if there exists an $\displaystyle \epsilon$-neighborhood $\displaystyle V_{\epsilon}(a)$ such that $\displaystyle V_{\epsilon}(a) \cap $A= $\displaystyle {a}$.

Any and all help would be greatly appreciated, thanks!