1)Let $\displaystyle S_1$ be a square in the coordinate plane. Divide this square into 4 equal squares by drawing lines straight down the middle. Pickanyone of the smaller squares, call it $\displaystyle S_2$. Now divide this square into 4 smaller squares, pick any one, call it $\displaystyle S_3$. And thus on. Let $\displaystyle s_1,s_2,s_3,...$ be the sequence of points which represent the centers of $\displaystyle S_1,S_2,S_3,...$ respectively. Show that $\displaystyle (s_n)$ convergences to some point.

2)Let $\displaystyle U$ be a subset of $\displaystyle \mathbb{R}$ which is closed under multiplication*. Let $\displaystyle S \mbox{ and }T$ two disjoint sets whose union is $\displaystyle U$. With the property that the product of any three elements is again in the set. Show that one of the sets $\displaystyle S,T$ must be closed under multiplication.

3)Let $\displaystyle x$ be a non-zero real number so that $\displaystyle x+x^{-1}$ is an integer. Show that $\displaystyle x^n + x^{-n}$ is an integer for every integer $\displaystyle n$.

*)Meaning, if $\displaystyle a,b\in U$ then $\displaystyle ab\in U$.