Proving: $A$ is closed iff $A = \bar{A}$.
"To the right": If $A$ is closed, $ A = \bar A$
If $A$ is closed this means that it contains all of its own accumulation points. And we would find that its derived set would be a subset of $A$. i.e. $ A' \subset A$ And we know that the closure of a set is equal to the union of its derived set and itself. So if $ A' \subset A$ then $ A' \cup A = A$ Therefore $\bar A = A$
"To the left": If $A = \bar A$ then $A$ is closed.
If $A = \bar A$ we would be saying that the set A is equal to the intersection of all closed super-sets of A. One of the properties of closed sets is that the intersection of any number of closed sets is closed. Therefore $A$ is closed.
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I wanted to show how I tend to prove things. Practicing proofs at random for a test I have tomorrow and hope to find out from you guys if my lengthy (wordy) approach to mathematical proofing is acceptable. This is a way that makes me feel confident in my understanding of the topics and is a lot more fulfilling for me than typing row after row of math symbols.
My other issue is citing previously taught theorems. How do I do that in such a way as to avoid using the numbering systems of different texts? One book may call that theorem, 2.1 and another 5.3. Most of our notes do not explicitly name theorems, so how do I use one theorem to prove another?
Seeing as I'm interested in doing a masters in mathematics next year I want to know if my way of looking at/explaining things or answering questions is acceptable at a higher level. I know at the moment my lecturers aren't always pleased by my "more paper please!" aproach to tests but I'd like to hear your opinions.