Recent content by Paparazzi

Again, the textbook has not mentioned sequences yet, so they cannot be used to solve the problem. I am also not looking for any other way to solve it other than directly, because I am fairly certain that it can be done so, and that is an exercise that I have created for myself to make sure that...

I just realized while walking from the library that nothing I typed made any sense whatsoever, since I made no hypothesis about n being a limit point. Actually I think it made sense, but had nothing to do with the question, and therefore made zero sense within the context of this question.

Oh I see. At this point in the textbook, he hasn't said anything other than the definition of a sequence itself. That's exactly what I was thinking as well. I do think I finally see where I was confused now. By vacuous truth, I showed that the set was closed, but that did nothing to show that...

I'm a little confused how that is relevant to a direct proof of showing that the set of all integers is closed. Edit: Actually, if the statement is vacuously true, then it actually shows nothing at all, since then we could conclude nothing about the set of all integers, correct? Serves me right...

When considered as a subset of \mathbb{R}^2, \mathbb{Z} is a closed set. Proof. We will show, by definition, that \mathbb{Z} \subset \mathbb{R}^2 is closed. That is, we need to show that, if n is a limit point of \mathbb{Z}, then n \in \mathbb{Z}. I think this becomes vacuously true, since our...

That is exactly what I needed. Thank you so much!

So since the statement is true for all \epsilon > 0, that's the reason you can just substitute in what is needed (since \epsilon/2 > 0)? Thanks a bunch for the reply.

Homework Statement Suppose the functions f and g satisfy the following property: for all \epsilon > 0 and all x, \text{if } 0 < | x - 2 | < \text{sin}^2(\frac{\epsilon^2}{9}) + \epsilon, \text{then } | f(x) - 2 | < \epsilon. \text{if } 0 < | x - 2 | < \epsilon^2, \text{then } | g(x) - 4 | <...

I need to write a paper on something to do with (general) topology, and we are encouraged to try and relate it to something that we enjoy. I really like probability (at least basic probability + stochastic processes), and I'm wondering if someone might suggest topic(s) that might be of interest...