Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

So, the representation of this is simple:

$${\{x \in \mathbb R | -1 \le x \le 1/n\}}$$

But I'm not sure what I need to prove. This is all of the information that I have. In class, a different problem, we were told what the set was equal to, and had to prove/disprove that. So, should I prove $\cup_{n = 1}^\infty [-1, 1/n] = [-1, 1]$?

Or is there something else that I need to do?

share|cite|improve this question
In your simple representation ${\{x \in \mathbb R | -1 \le x \le 1/n\}}$, what is the value of $n$? – MJD Oct 18 '12 at 3:29

You’ll have a hard time proving that $$\bigcup_{n = 1}^\infty \left[-1, \frac1n\right] = \left[-1, \frac1n\right]\;:$$ it doesn’t even make sense. The $n$ on the lefthand side is a dummy variable: the value of the expression wouldn't change if you replaced it by something else, say $k$, to get $$\bigcup_{k = 1}^\infty \left[-1, \frac1k\right]\;.$$ The $n$ on the righthand side, however, is apparently a particular integer. Thus, you’re using one letter, $n$, to represent two unrelated things of very different kinds.

You’re actually being asked to figure out exactly what real numbers are in the set $$\bigcup_{n = 1}^\infty \left[-1, \frac1n\right]$$ and then to express the set in a way that doesn’t require talking about infinitely many sets. For example, it should be clear that every real number in the interval $[-1,0]$ belongs to the set. However, $$\bigcup_{n = 1}^\infty \left[-1, \frac1n\right]\ne[-1,0]\;,$$ because, for instance, $$\frac13\in\bigcup_{n = 1}^\infty \left[-1, \frac1n\right]\;,$$ but $\frac13\notin[-1,0]$.

Sketch the intervals $\bigcup_{n = 1}^\infty \left[-1, \frac1n\right]$ for the first few positive integers $n$ to get an idea of what they look like. Once you’ve done that, figure out exactly what the set $$\bigcup_{n = 1}^\infty \left[-1, \frac1n\right]$$ looks like, and write down a simple description of that set. Finally, if your set is $A$, prove that $$\bigcup_{n = 1}^\infty \left[-1, \frac1n\right]=A\;,$$ probably by showing that if $x\in A$, then $x\in\bigcup_{n = 1}^\infty \left[-1, \frac1n\right]$, and if$x\in\bigcup_{n = 1}^\infty \left[-1, \frac1n\right]$, then $x\in A$.

share|cite|improve this answer

I think you're asked to represent the infinite union in a closed way:

$$\bigcup_{n=1}^\infty \left[-1\,,\,\frac{1}{n}\right]=[-1\,,\,1]$$

Can you see why?

share|cite|improve this answer
Yep, thank you. – DonAntonio Oct 18 '12 at 3:31
Can't understand why someone would downvote an obvious typo. At least I can cancel it! – André Nicolas Oct 18 '12 at 3:33
Thanks @André. I don't get very annoyed by this any more. Some people just seem to enjoy downvoting, so let knock themselves out. – DonAntonio Oct 18 '12 at 3:37

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.