Consider a set . ( can be finite or infinite)
Let be subsets of , such that
2. is finite
Prove that there exits , such that for any ,
I want to prove this in a rigorous way. Here is my attempt
1. Claim there exists an , such = . If this claim is true proving above is trivial.
2. So to prove the claim. Consider any . Thus must belong to some . Let be minimum such . Iterate over all the elements of (say ' ' elements, as is finite). We will have a set . Now claim that . (This claim is easy to prove)
I have two questions here:
1. Is the above proof rigorous enough !
2. How do I prove that exists? More generally if is a finite subset of , prove that exits and is unique?
Please help !