Line and Line segment will be used interchangeably. For the sake of clarity line will be used instead of line segment as it will be later proven lines are indeed line segments.
1. A line is composed of further lines. This can be observed within the standard number line.
2. Each line as composed of lines is thus a set of lines. Each 1 line as composed of multiple lines necessitates that each line, as a line in itself, contains itself.
3. Each set contains a set, thus each set is a fractal of another set as each fraction of a number line is both a fractal of the line and set of whole numbers in itself.
4. Sets replicate sets recursively, thus each set is a fractal/fraction. This is evidenced by the number line.
5. An empty set, that which contains nothing, is effectively the line itself again. As each unit interval (line) is directed from point 0 to another point zero, the line effectively is a 1 dimensional loop as it returns to its origins. Its beginning is a 0d point, its end is a 0d point.
6. Point 5 is important as it reflects back to point 2. That line, or unit interval, as composed of fractions/fractals (which in themselves are whole numbers as evidence by the number line being hold numbers that are fractions of one line) observes the line in beginning and end at the same point 0 perfectly is a loop which contains these fractions/fractals (which again are whole lines and numbers in themselves).
7. Each line as projecting from and back to its origins of 0 (0d point), exists as a loop contained by another loop. Each loop is intrinsically empty, containing other intrinsically empty loops. Thus each set as empty, is a set containing empty loops.
A pictorial description would be a loop containing loops is always empty as the loops are empty.
8. Each set is thus a fractal, as each line is a fractal of another line. This fractal within fractal, continually projects ad-infinitum. Thus this in itself is a line again. An infinite line, projecting away from its center point of zero is projecting towards 0 as the line segment which reflects it does the same thing. A line segment composed of infinite numbers, necessitates a projection to point 0.
Thus the infinite line and line segment are the same.