Question

Given that H is a set and that x is an interior point of the set R∖H, prove that x is not close to H.

So, I have an example where they are opposite of the question, I just don't know how to spin it around.

Suppose that S is a set of real numbers and that x is a given number. Then the following two conditions are equivalent to one another:

1.The number x is close to the set S.
2.The number x is not an interior point of the set R∖S.

Proof that Condition 1 Implies Condition 2
We assume that x is close to S.
For every δ>0, the interval (x-δ,x+δ) must contain a member of S.
For every δ>0, it is impossible for (x-δ,x+δ) to be a subset of R∖S.
We conclude that x can't be an interior point of R∖S.
Proof that Condition 2 Implies Condition 1
We assume Condition 2.
For every δ>0 we are guaranteed that (x-δ,x+δ) is not a subset of R∖S.
For every δ>0, we are guaranteed that (x-δ,x+δ)∩S≠∅.

Answer 1

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