1)Let be a square in the coordinate plane. Divide this square into 4 equal squares by drawing lines straight down the middle. Pick any one of the smaller squares, call it . Now divide this square into 4 smaller squares, pick any one, call it . And thus on. Let be the sequence of points which represent the centers of respectively. Show that convergences to some point.
2)Let be a subset of which is closed under multiplication*. Let two disjoint sets whose union is . With the property that the product of any three elements is again in the set. Show that one of the sets must be closed under multiplication.
3)Let be a non-zero real number so that is an integer. Show that is an integer for every integer .
*)Meaning, if then .