A Readable Introduction to Real Mathematics by Daniel Rosenthal & David Rosenthal & Peter Rosenthal

Author:Daniel Rosenthal & David Rosenthal & Peter Rosenthal
Language: eng
Format: epub
ISBN: 9783030006327
Publisher: Springer International Publishing

There is a theorem that can often be used to provide very easy proofs that sets are countable. The next several results form the basis for that theorem and are useful in other contexts as well.

Theorem 10.3.9

A subset of a countable set is countable.

Proof

Let be a countable set. If is finite, then the result is clear. If is infinite, then there exists a one-to-one function, say f, mapping the set of natural numbers onto . Thus, the elements of can be listed in a sequence, (f(1), f(2), f(3), f(4), …). If is a subset of , then the elements of correspond to some of the elements in the sequence. Therefore, the elements of can also be listed in a sequence, and hence is either finite or has the same cardinality as . □

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