Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top
  1. The set $S$ of all rational numbers whose denominators are prime.

  2. $S = \{(-1)^{n}+\frac{1}{n}\mid n\in\mathbb{N}\}$

  3. $S = (0,1) $

I have answers for these from the back of a book but I'm not sure about the intuition behind finding the limit points.

Edit: I understand #3 now, all $x\in[0,1]$ are accumulation points since an open ball around any of those points contains infinitely many points of $S$.

Not sure on how to approach #1 though.

share|cite|improve this question
This community could help you better if you could explain your own thoughts on the problem and where you feel you are stuck. – Grumpy Parsnip Jan 24 '12 at 4:06

The formal definition states that a point $x$ is a limit point of $S$ unlesss there exists a neighborhood of $x$ that doesn't contain any points of $S$ (except, possibly, $x$ itself). Intuitively, if there are elements of $S$ that are really close to $x$, then $x$ is a limit point.

Let's handle your examples one at a time, starting with #3. The point $0$ is a limit point of $S$, because there are points in the open interval $(0,1)$ that are extremely close to $0$. Similarly, $1$ is a limit point, along with everything in $S$ itself. However, $17$ for example is not a limit point, because the neighborhood $(16, 18)$ contains nothing from $S$. This should be intuitively clear.

#2 is more difficult (for some reason, your textbook seems to put the harder problems first!), but you should be able to handle it if you consider the cases where $n$ is even and odd separately. Graph the points if you aren't sure.

#1 is less clear still. Hint: you should be able to show that $S$ is dense in $R$. You might want to use the fact that there are infinitely many primes.

Comment if you want any more help!

share|cite|improve this answer
By the way, "unlesss" means "unless and only unless". It's analogous to "iff". I have no idea how standard this notation is, but it's cute. – Lopsy Jan 24 '12 at 4:13

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.