- #1
avalle
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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 [itex]\in[/itex] O there exists a ε-neighborhood [itex]_{V}[/itex]ε (a) [itex]\subseteq[/itex]O.
Then I would use pointwise convergence to prove that for each n[itex]\in[/itex] N, let fn be a function defined on a set A [itex]\subseteq[/itex] 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).
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 [itex]\in[/itex] O there exists a ε-neighborhood [itex]_{V}[/itex]ε (a) [itex]\subseteq[/itex]O.
Then I would use pointwise convergence to prove that for each n[itex]\in[/itex] N, let fn be a function defined on a set A [itex]\subseteq[/itex] 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).