Recent content by zbr

There is an issue with the example you provided, it does not satisfy the property that xm ≠ xn for m ≠ n, so that example doesn't work for this question... so I believe the proof that I gave earlier is the correct way to procede with the first part of that question.

I would agree with you that 2 is not a projected cluster point and that (2,0) is a cluster point of the sequence... but wouldn't (2,0) be a cluster point using any disk, regardless of it being punctured or not? The definition that we have been given in class of cluster points is as such: x in...

Ok here is what I have: Assume that (a,b) is a cluster point with one of a or b not being 0 or 1. Then we have a A ∩ Br(a,b)(punctured) ≠ ∅ for all r. This implies that Ax ∩ Br(a)(punctured) ≠ ∅ and Ay ∩ Br(b)(punctured) ≠ ∅ which implies that x is a cluster point of Ax and b is a cluster...

Yes, ok so here the cluster points of the projections would be {0,1} for both projections. The cluster points of the sequence then would be (0,1) and (1,0) but not (1,1) or (0,0). Ok so that clarify's the equality situation, thanks! However, for the roof I am still a bit stumped...

My apologies for making a confusing example. If I am understanding you correctly, I believe you mean the following sequence: (0,1), (1,1), (1,0), (1/2,1/2), (0,1), (1/3,1/3), (1,0), (1/4,1/4), ... So here the cluster points of the projections would be 0 for both cases. But wouldn't (0,0) be...

Well here i am thinking: (0,1), (0,2), (1,0), (2,0), (0,1/2), (0,3/2), (1/2,0), (3/2,0), (0,1/3), (0,4/3), (1/3,0), (4/3,0)... In this cae both the x and y projections would have cluster points at 0 and 1, and here I'm pretty sure that (0,0) is a cluster point, but then the point (0,1)...

Thanks Dick for the reply, In this case (0,0) would not be a cluster point of the sequence, but I would also argue that 0 nor 1 is a cluster point of the projections. This is because if we consider the projection on the x-axis: Ax ∩ B0.5(0)(punctured) is empty since we have a punctured ball...

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...

Ah, yes of coarse, set sequence 1, 1/2, 1/4, 3/4, 1/8, 3/8, 5/8, 7/8, 1/16, ... would satisfy the sequence with convergent subsequences within the interval [0,1], then just throwing in divergent terms would satisfy the second requirement. Thanks so much for your help Dick!

Hello all, (*) I have a question about convergent subsequences. Specifically I am looking for an example of a sequence that is unbounded but who has convergent subsequences in the interval [0,1]. A similar question would be to have an unbounded sequence, but who has a convergent...

Hello all, I am having trouble with a convergent series problem. The problem statement: Let f:ℝ→ℝ be a function such that there exists a constant 0<c<1 for which: |f(x)-f(y)| ≤c|x-y| for every x,y in ℝ. Prove that there exists a unique a in ℝ such that f(a) = a. There is a provided hint...