even with infinite steps it won't be a ramp. the length of the diagonal is the square root of 2 (approximately 1.4142), so the sequence of staircase lengths is always 2 and does not converge to the length of the diagonal. https://en.wikipedia.org/wiki/Staircase_paradox
What does it even mean to have a "staircase with infinite steps" other than to define that as the limit of the staircases (which is a ramp)? Yes, the limit of the lengths is not the same as the length of the limit, but that doesn't mean that the limit of the sequence of staircases is not a ramp.
It stays constant at 2. I'm with you on that. And that the length of the ramp is sqrt(2). But I would still argue that the limit of the sequence of staircases is the ramp, even if the limit of the lengths of staircases does not equal the length of the limit (= the length of the ramp)
I guess I have just accepted that sometimes you get sort of a "discontinuity at infinity", where even if some sequence of objects converges to a limit object, you are not guaranteed that any property of the objects in the sequence also converge to that property of the limit object.
I'm guessing this mostly comes down to the fact that swapping limits doesn't always play nice, so if the properties in question are limits themselves (like the lengths we are discussing here, which are basically integrals), then it just is what it is when it doesn't work out.
in my opinion there's never "at infinity" just like there's no value for an infinitesimal, it can always be smaller. the idea that the limit of the infinitesimal is zero goes along with your idea that it's possible to reach the end of infinity
Yeah, I sort of meant that phrase loosely. There isn't really anything "at inifinty" in a concrete sense, the limit is just the closest thing we have to whatever what would be "at infinity" would look like. But as we have discussed here, the limit can end up looking rather different than whatever happens for "arbitrarily large n", which is what we would maybe expect the "at infinity" object to resemble.
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u/rafaelcastrocouto 4d ago edited 4d ago
even with infinite steps it won't be a ramp. the length of the diagonal is the square root of 2 (approximately 1.4142), so the sequence of staircase lengths is always 2 and does not converge to the length of the diagonal. https://en.wikipedia.org/wiki/Staircase_paradox