Prove that a finite union of countable sets is countable. Therefore, if A is uncountable and B ⊂ A is countable, then A \ B is uncountable.

First of all we have to prove that,

The union of countable family of countable sets is countable.

PROOF:

Without loss of generality, we can denote a countable family of sets by $A_1, A_2,A_3\cdots \cdots \cdots \cdots$

Suppose $a_{i1}, a_{i2}, a_{i3},\dots \dots \dots$ is an enumeration for $A_i$. Then,

$\underset{i=1}{\overset{\infty }{Y}} A_i=\{ a_{ij} : i\in \mathrm{\mathbb{N}}, j\in \mathrm{\mathbb{N}}\}$

To the element $a_{ij}$ we assign a natural number $n$ so that there corresponds at most $n-1$ distinct elements of A.

Therefore, A is countable by Countable Lemma.