Set of Points on Line Segment is Infinite
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The set of points on a line segment is infinite.
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.
- 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?