>>5948823>>5948900>>5948977I invented a very similar paradox to this, that I would pose to people in math forums and IRL hangouts with math-minded people to see if they could respond with a satisfactory explanation.
https://files.catbox.moe/ch4v1z.PNGThe hypotenuse of a right triangle with leg lengths 3 and 4 is known to have length 5; it's a very famous "3-4-5 right triangle" and the result of the Pythagorean Theorem. Therefore, the line connecting the points (3, 0) and (0, 4) in the 2D coordinate plane will always have a length of 5.
But consider a similar "triangle" where instead of a straight line from (3, 0) to (0, 4), you have a jagged line that makes small movements horizontally and vertically in between the two points. What is the length of this "staircase line" connecting the same two points, with distances 3 and 4 meeting at a right angle? Well, the horizontal lengths of the staircase will always add up to 3, and the vertical lines of the staircase will always add up to 4. That means the length of this line will always be 7, even though if you continue this process indefinitely then the line should infinitesimally approach the hypotenuse line.
Where does this break down? From a more thorough and rigorous understanding of Calculus actually, which is why in college level math you study things like Real Analysis, which is a more rigorous foundational approach to Calculus. It's true that the staircase line will always have length 7, and it's also true that you can make the staircase line as infinitely close to the hypotenuse line as possible. But the problem is that there is no length-preserving convergence. It's true that infinite series like 1/1 + 1/2 + 1/4 + 1/8 + 1/16 ... equals 2 by convergence, because the expression on the left and the value on the right can be made infinitesimally close to each other in value. But as you infinitesimally smoothen the staircase to approach the hypotenuse, you are always consistently maintaining the length value 7 of that line - you can make a statement of convergence on the 2D shape of the line, but not the length. The jaggedness of the line does play into this; if you had a different type of convergent transformation into the hypotenuse line that created a continuous and smooth line, then not only would you approach the hypotenuse in shape but you would also approach it in length, in which case you could say that the end result of the convergence actually *is* the actual hypotenuse line with length 5.
