I presented a professor with the following proof:
Prove that the empty set is closed.
By definition : is closed <=> cl <=> ( cl => )
cl is the closure of the empty set,and
B(x,r) is a ball of radius r round x
But ,by definition again cl <=> for all r>0 ,B(x,r) .................................................. ..................1
But , B(x,r) => (B(x,r) or ) <=>
(B(x,r) => )
And using (1) we get :
Thus ,we have proved:
( cl => )
And the empty set is closed.
The professor did not accept the proof as correct .
Do you agree with him??
And that's what makes it a bad proof. I know I'll get flack for this but I'm telling you that a proof is not just composed of it's content, it's about its efficiency and coherency.
Usually professors are confused with this kind of proof
The mistake is when you say "And using (1), we get ". Not because this conclusion is false or nonsense but just because your apparent logic is flawed.
Indeed, you write correctly (for , but works in general):
(if you look at it right in the eye, b) is somewhat funny)
From there, you deduce , don't you? Then you've just proved that for any set .
This was an illustration of my second point: obfuscation. You are allowed to write very formal proofs, but that requires skill and insight (both from you and your reader); if you perform blind logical manipulation, you are very likely to end up writing junk, to say the least. Furthermore, maths is not just about writing correct statements that follow logically from one another; it is mainly about ideas, constructions, insights. And writing a proof is usually about wanting it to be understood, not just checked for validity. It is important that proofs are correct, but a good mathematician is also one who knows when an argument is missing without having to write the proof up to every single logic axiom involved.
A correct proof along your lines could be (according to me): if there exists , then in particular , which rewrites as (the left-hand side being a subset of the empty set). This conclusion is false, hence the assumption was false as well. Thus there is no element in and we have .
If you see the closure of a set (in a metric space as well) as the set of the limits of convergent sequences with values inside this set then, as Plato said, the conclusion is also straightforward: the set of the sequences inside the empty set is empty hence the set of their limits is empty as well.