Finitely generated implies countable

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., finitely generated group) must also satisfy the second group property (i.e., countable group)
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Statement

Any finitely generated group is a countable group.

Proof

We can explicitly enumerate, in a countable fashion, all possible words on a finite generating set. Since this provides a listing of all elements of the group (possibly with repetitions) the group is countable.