Question P.167: Consider the line segments in the xy-plane formed by connect......
Consider the line segments in the xy-plane formed by connecting points on the positive x-axis with x an integer to points on the positive y-axis with y an integer. We call a point in the first quadrant an I-point if it is the intersection of two such line segments. We call a point an L-point if there is a sequence of distinct I-points whose limit is the given point. Prove or disprove: If (x, y) is an L-point, then either x or y (or both) is an integer.
The given statement is true: If (x, y) is an L-point, then either
x or y is an integer.
Suppose (\bar{x},\bar{ y}) is a point in the first quadrant with neither \bar{x}~nor~\bar{y} an integer. We will show that (\bar{x},\bar{ y}) is not an L-point. Choose positive integers M and N for which the line between (M, 0) and (0, N) passes above the point (\bar{x},\bar{ y}). Let ε be any positive number less than all three of the following: (i) the distance from the above line to (\bar{x},\bar{ y}), (ii) the distance from \bar{x} to the closest integer, and (iii) the distance from \bar{y} to the closest integer.
We claim that only finitely many lines whose x- and y-intercepts are both positive integers can pass within ε of (\bar{x},\bar{ y}). To see this, suppose m and n are positive integers for which the line between (m, 0) and (0, n) passes within ε of (\bar{x},\bar{ y}). Then condition (i) above implies that either m < M or n < N.
Given m, consider the two tangent lines to the circle with center (\bar{x},\bar{ y}) and radius ε which pass through (m, 0). Since the line connecting (m, 0) to (0, n) must lie between these tangent lines, we see from condition (ii) above that there are at most finitely many possibilities for n. Similarly, condition (iii) implies that given n, there are at most finitely many possibilities for m. So, indeed, only finitely many lines whose x- and y-intercepts are both positive integers pass within ε of (\bar{x},\bar{ y}). Since every I-point within ε of (\bar{x},\bar{ y}) has to be one of the finitely many intersections of these lines, the point (\bar{x},\bar{ y}) cannot be an L-point, hence the statement is true.
