I tried as follows:
If x is not in the interior of set E, then no neighborhood of x is contained in E. So for every neighborhood of x there is a point y which is not in E. That is, every neighborhood of x contains a point y which lies in complement of E. So x is a limit point of complement of E.
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On the other hand, if y is in closure of complement of E, then y is in E complement or set of all limit points of it. In the first case we are done since Int(E) is contained in E.
In the second case, y is the limit point of complement of E. So every neighborhood of y contains a point which is not in E. Hence no neighborhood of y completely lies in E. Thus y is not an interior point of E.
Is this correct?? Thanks in advance.
Let X be discrete, E = X - {a}. Assume x in interior E.
In your 1st proof, you conclude x is a limit point of the
complement of E, namely {a} which has no limit points.
That example shows your proof is wrong.
You do however, have the right idea.
Use the right definition of closure:
the set of all adherence points.
a is an adherance point of A when for all
open U nhood a, U $\cap$ A is not empty.