Prove that if A is denumerable and B is a finite subset of A, then A \ B is denumerable.
Lets assume you are using the definition of denumerable as meaningOriginally Posted by mathkid
countably infinite.
Let be an enumeration of .
We can rearrange this enumeration so that <you may need to prove that this can be done>:
( is the number of elements of )
Then:
.
So where is an enumeration of
RonL
I like to use the definition of "countable" as "can be injected into the natural numbers". It is then immediate that a subset of a countable set is countable. "Denumerable" is countable and not finite, where "finite" means every injection of the set to itself is also a surjection. All that remains is to prove that the union of two finite sets is finite; use the lemma that a finite set injects into {1..n} for some natural number n.