Set of Points on Line Segment is Infinite

Theorem

The set of points on a line segment is infinite.

Proof

Let $S$ denote the set of points on a line segment.

Aiming for a contradiction, suppose $S$ is finite.

Then there exists $n \in \N$ such that $S$ has $n$ elements.

Let $s_1$ and $s_2$ be two arbitrary adjacent points in $S$.

That is, such that there are no points in $S$ between $s_1$ and $s_2$.

But there exists (at least) one point on the line segment between $s_1$ and $s_2$ which is not in $S$.

Hence there must be more than $n$ elements of $S$.

From that contradiction it follows by Proof by Contradiction that $S$ is not finite.

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 1$ What is infinity?