- #1
zbr
- 11
- 0
Hello all,
I am having trouble with a homework problem. The problem is as such:
Let a = {zn = (xn,yn) be a subset of ℝ2 and zn be a sequence in ℝ2 such that xn ≠ xm and yn ≠ ym for n≠m.
Let Ax and Ay be the projections onto the x and y-axis (i.e. Ax = {xn} and Ay = {yn}. Assume that the cluster points of Ax = {0,1} and the cluster points of Ay = {0,1}.
a) Prove that the cluster points of A (denoted A') are a subset of {(0,0), (0,1), (1,0), (1,1)}.
b) Can A' = ∅? If yes, give an example; if no provide a proof.
c) Assume that A is bounded. Can A' consist of a single point? If yes, give an example; if no provide a proof.
Here is what I have so far:
a) Here I am confused as to why this statement is not an equality. For example, if we consider the point (0,0), then for every r A∩Br(punctured) is non empty since 0 is a cluster point of both the x and y projections. Furthermore, shouldn't this set include more points i.e. (1,y) where y is in yn? I am most likely thinking of this wrong... and my confusion here is preventing me from attempting the other two parts of the problem. Any and all help is much appreciated!
I am having trouble with a homework problem. The problem is as such:
Let a = {zn = (xn,yn) be a subset of ℝ2 and zn be a sequence in ℝ2 such that xn ≠ xm and yn ≠ ym for n≠m.
Let Ax and Ay be the projections onto the x and y-axis (i.e. Ax = {xn} and Ay = {yn}. Assume that the cluster points of Ax = {0,1} and the cluster points of Ay = {0,1}.
a) Prove that the cluster points of A (denoted A') are a subset of {(0,0), (0,1), (1,0), (1,1)}.
b) Can A' = ∅? If yes, give an example; if no provide a proof.
c) Assume that A is bounded. Can A' consist of a single point? If yes, give an example; if no provide a proof.
Here is what I have so far:
a) Here I am confused as to why this statement is not an equality. For example, if we consider the point (0,0), then for every r A∩Br(punctured) is non empty since 0 is a cluster point of both the x and y projections. Furthermore, shouldn't this set include more points i.e. (1,y) where y is in yn? I am most likely thinking of this wrong... and my confusion here is preventing me from attempting the other two parts of the problem. Any and all help is much appreciated!