Thread: What are the limit points of this set?

1. What are the limit points of this set?

What are the limit points of the set containing x^n such that n is in the positive natural numbers for each a that is a member of the inteval [-1,1]?

2. Originally Posted by kevinlightman
What are the limit points of the set containing x^n such that n is in the positive natural numbers for each a that is a member of the inteval [-1,1]?

$\displaystyle x^n=1 \Longleftrightarrow x=1\,\,\,or\,\,\,x=-1\,\,and\,\,n\,\,even\,,\,\,x^n=-1\,\,if\,\,x=-1\,\,and$ $\displaystyle n\,\,odd\,,\,\,and\,\,\,x^n\rightarrow 0\,\,\,if\,\,\,|x|<1$

Now prove the above and you get your answer.

Tonio

3. Originally Posted by tonio
$\displaystyle x^n=1 \Longleftrightarrow x=1\,\,\,or\,\,\,x=-1\,\,and\,\,n\,\,even\,,\,\,x^n=-1\,\,if\,\,x=-1\,\,and$ $\displaystyle n\,\,odd\,,\,\,and\,\,\,x^n\rightarrow 0\,\,\,if\,\,\,|x|<1$

Now prove the above and you get your answer.

Tonio
I have already done this, it is more the concept of limit points I don't understand, I believe 0 to be one for every value not equal to -1 1 or 1 but I can't see what the others would be

4. Originally Posted by kevinlightman
I have already done this, it is more the concept of limit points I don't understand, I believe 0 to be one for every value not equal to -1 1 or 1 but I can't see what the others would be
Tell me, what is a limit point? Why do you think that $\displaystyle 0$ is a limit point of this set?