Lindelöf's Lemma/Lemma 2
Lemma
Let $R$ be a set of real intervals with rational numbers as endpoints.
Let every interval in $R$ be of the same type of which there are four: $\openint \ldots \ldots$, $\closedint \ldots \ldots$, $\hointr \ldots \ldots$, and $\hointl \ldots \ldots$.
Then $R$ is countable.
Proof
By Rational Numbers are Countably Infinite, the rationals are countable.
By Subset of Countably Infinite Set is Countable, a subset of the rationals is countable.
The endpoint of an interval in $R$ is characterized by a rational number as every interval in $R$ is of the same type.
Therefore, the set consisting of the left hand endpoints of every interval in $R$ is countable.
Also, the set consisting of the right hand endpoints of every interval in $R$ is countable.
The cartesian product of countable sets is countable.
Therefore, the cartesian product of the sets consisting of the respectively left hand and right hand endpoints of every interval in $R$ is countable.
A subset of this cartesian product is in one-to-one correspondence with $R$.
This subset is countable by Subset of Countably Infinite Set is Countable.
By Set which is Equivalent to Countable Set is Countable:
- $R$ is countable.
$\blacksquare$