Sequence that converges to a point

[avalle]

1.Problem Statement:

If O is an open subset of ℝ does there exist a sequence in O that converges to x? Explain.

2.Relevant equations

3. The Attempt at a Solution

So if I define a open subset of ℝ to be open if for all points x \in O there exists a ε-neighborhood _{V}ε (a) \subseteqO.

Then I would use pointwise convergence to prove that for each n\in N, let fn be a function defined on a set A \subseteq ℝ. The sequence fn of function converges pointwise on A to a function f : A → ℝ if for all x in A the sequence of real numbers fn(x) converges to f(x).

 

[Gengar]

You need to consider 3 cases:
1. x is in O
2. x is a boundary point of O
3. x is neither in O nor a boundary point of O

Should be straightforward from there