how to find limit points Which of the followings is correct:
Let S$\subseteq$ $\mathbb R$$^2$ be defined by
S={(m+$\LARGE$1/$2^{\mid p \mid}$ ,n+$\LARGE$1/$2^{\mid q \mid}$ : m,n,p,q $\in$ $\mathbb Z$}. Then,
1.S is discrete in $\mathbb R$$^2$.


*The set of limit points of S is the set {(m,n):m,n $\in$ $\mathbb Z$}.


Need some help please.

If S is discrete in $\mathbb R^2$, then every point x has a neighborhood U such that $U\cap S = \{x\}$. In particular the point (1,1) (which is $(0+1/2^0, 0+1/2^0))$ has a neighborhood with no other points of S in it. However, 
$$(1+\frac{1}{2^n},1+\frac{1}{2^n})$$
is in S for all non-negative integers n, which implies that no neighborhood of (1,1) can exclude all other memebers of S.
If the set of limit points is $ \{(m,n):m,n\in\mathbb Z\}$ this is equivalent to two statements: Every point (m,n) with $m,n\in\mathbb Z$ is a limit point, and every limit point is of the form (m,n) with $m,n\in\mathbb Z$.
The first statement can be proven the same way as point 1 was. However, the second is not true because we can take the series of points
$$(1+\frac{1}{2^n},\frac{1}{2})$$
and show that $(1,\frac{1}{2})$ is a limit point not of the required form.
