How can you prove that the limit of a sequence is a limit point of a set?

[dancergirlie]

Homework Statement

Let x \in R, and let A \subset R. Let (an) be a sequence with an \in A and (an) \neq x for all n \in N, and assume
that x = lim (an.) Prove that x is a limit point of A.

Homework Equations

The Attempt at a Solution

Suppose that x=lim(an). Meaning there exists a N \in N so that for n\geqN:
|an-x|<\epsilon

Which would make the \epsilon- neighborhood of (an)= (an- \epsilon, an+ \epsilon)

After this I don't know what to do. I need to show that every epsilon neighborhood of V(x) intersects A in some point other than x.

Any tips would be great!

 

[snipez90]

Careful, your e-neighborhood is off. Remember |a_n - x| < e means x - e < a_n < x + e or a_n is in (x - e, x + e). Now you know that a_n =/= x for each n, so what can you conclude.

Of course this proposition is an if and only if, but this direction is probably clearer since "all but a finitely many points" certainly implies "at least one".

 

[dancergirlie]

alright so if the epsilon neighborhood is (x-e,x+e) and (an) is unequal to x for all n in N would that mean that for some N in N when n>=N that (an) is contained in the epsilon neighborhood. Which would mean that x is the limit point of (an)?

 

[dancergirlie]

oh wait, nevermind, i think I got it. I was trying to make it more complicated than it really was. I am finding this whole open sets, closed sets and limit points idea kind of confusing.