1- if A1,A2,A3,...,Am are each countable sets, then the union A1 U A2 U A3...U
Am is countable
Proof. We only need to show the statement is true for two countable sets.
Statement: If A and B are two countable sets, then the union of A and B is a
countable.
Let A = {an | n ∈ } and B = {bn | n ∈
{c n } n 1 as follows.
} be countable. We construct a sequence
c2k = ak and c2k + 1 = bk for all k ∈ .
Then A ∪ B = {cn | n ∈ } and since it is infinite, it is countable.
By the statement, we know that if A1,A2,A3,...,Am are each countable sets, then the
union A1 U A2 U A3...U Am is countable . Since A1 and A2 are countable, so is
A1 ∪A2. So is A 1 ∪ A2 ∪ A3,…; so is A 1 ∪ A2 ∪ A3 ∪… ∪ Am.

2- If An is a countable set for each n belong to N, then
A
n
is countable.
n 0
Proof. We can use the above graph to prove this statement. If we count every element
from A0, A1,A2,…, as direction as arrow in the graph, we can count all elements in all
Ai’s. So, it is countable. This technique is called the breath-first-search method