proof that countable unions are countable
Let be the set of positive primes (http://planetmath.org/Prime). is countably infinite, so there is a bijection between and . Since there is a bijection between and a subset of , there must in turn be a one-to-one function .
Each is countable, so there exists a bijection between and some subset of . Call this function , and define a new function such that for all ,
We may now define a one-to-one function , where, for each , for some where (the choice of is irrelevant, so long as it contains ). Since the range of is a subset of , is a bijection into that set and hence is countable.
|Title||proof that countable unions are countable|
|Date of creation||2013-03-22 12:00:01|
|Last modified on||2013-03-22 12:00:01|
|Last modified by||Koro (127)|