Question

find the set of limit points of

Answer 1

R={ x/x∈R }

Answer 2

@Mchilds15

Answer 3

@zepdrix

Answer 4

@Callisto

Answer 5

@texaschic101 @beccaboo333 @mathslover

Answer 6

@johnweldon1993 @chmvijay @uri

Answer 7

@Chris911

Answer 8

@Ashleyisakitty

Answer 9

@inkyvoyd

Answer 10

There are no limit points. For any neighborhood of \(x\in R\), you will never find a point that does not lie in \(R\).

Answer 11

can you tell me stepwise,i have to explain the procedure of how it is solved

Answer 12

^ that implies that every point is a limit point, not that there are none.

Answer 13

how to prove it @Jemurray3

Answer 14

What's the definition of a limit point?

Answer 15

@Jemurray3, you're right, I had the definition wrong in my mind.

Answer 16

A point x in X is a limit point of S if every neighbourhood of x contains at least one point of S different from x itself?

Answer 17

Yes. So, written in more precise language,

A point \( x \in S\) is a limit point of \(S\) if, for all \( \epsilon > 0\), there exists a point \(y \in S\) such that \( 0 < |x-y| < \epsilon \).

So a proof that every point in the set of real numbers is a limit point might begin like this:

Let \( \epsilon > 0\) and \(x \in R\) be given. We seek to demonstrate that there exists some \( y \in R\) such that \( 0 <|x-y| < \epsilon \). [...]

So your job now is to pick some real number Y that would make that statement true. Don't think too hard - there are an infinite number of choices, and all of them are fairly obvious.

Answer 18

oh thanks ^^,