I have a four part question to which I can only seem to answer a few parts.
(i) Prove that each open interval is a set in and each closed interval as well.
(ii) Prove that each open and closed interval is an set.
(iii) Prove that is an set.
(iv) Prove that the compliment of an set is an set and vice versa.
In order to prove (i) I thought of a union of closed intervals that expanded but were limited between a and b. Since all real numbers have a sequence with itself as a limit, Cauchy, I thought about taking the monotonically decreasing subsequence and monotonically increasing subsequence of each of them to form an expanded interval. For instance:
There exists and monotonically increasing sequence and a monotonically decreasing sequence .
To prove that [a,b] is the union of a closed interval just make take a cousin cover of that interval.
For (ii) I've got no idea how to make the open interval but the closed interval is easy as it can be the intersection of two overlapping open intervals.
On (iii) and (iv) I've got no clue.
Could someone tell me if what I have done is right and give me a few hints as to how to solve the remaining portions?