Lindelöf's Lemma/Lemma 2

Theorem
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: $\left({\ldots \,.\,.\, \ldots}\right)$, $\left[{\ldots \,.\,.\, \ldots}\right]$, $\left[{\ldots \,.\,.\, \ldots}\right)$, and $\left({\ldots \,.\,.\, \ldots}\right]$.

Then $R$ is countable.

Lemma
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.

$R$ is countable by Lemma as $R$ is in one-to-one correspondence with a countable set.