# Lindelöf's Lemma/Lemma/Lemma

## 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

### Lemma 2

Let $S$ be countable set.

Let $T$ be a set.

Let $T$ be in one-to-one correspondence with $S$.

Then $T$ is countable.

$\Box$

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

$\blacksquare$