Show that a subsct of countable set is countable

We can prove this theorem as follows:

Step-by-step explanation:

Let where is a countable set.  So we can arrange the elements of as  a sequence we have the following cases:

1. A is finite. If A is finite clearly it is a countable set.

2. A is infinite. Define the sequence of positive integers as follows:

is the less positive integer such that

is the less positive integer grater than such that .

Now, observe that the correspondence is a one-to-one correspondence between the elements of and . So, is   a countable set.

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