Question

Show that a subsct of countable set is countable

Answer 1

Answer:

We can prove this theorem as follows:

Step-by-step 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.