Question

Show that a subsct of countable set is countable

Answer 1

We can prove this theorem as follows:

Explanation:

Let
`A\subseteq S` where
`S` is a countable set. So we can arrange the elements of
`S` as a sequence
`S=\{x_(n)\}_(n=1)^(\infty )` 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
`\{n_i\}_(i=1)^(\infty)` as follows:

`n_(1)` is the less positive integer such that
`x_{n_(1)}\in A`

`n_(i)` is the less positive integer grater than
`n_(i-1)` such that
`x_{n_(i)}\in A.`.

Now, observe that the correspondence
`f(i)=x_(n_i),\,\,(i=1,2,3,...)` is a one-to-one correspondence between the elements of
`A` and
`\mathbb{N}`. So,
`A` is a countable set.