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) $\subseteq$O.

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