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What is it time for? No idea.
Okay, it's time for convergence, as that title says. Why is it time for convergence?
The answer is simple. The first question of your homework is on it.
I assume I am violating 300 copyright laws by doing this. If you work at the Mathematical Insitute in Oxford University, please don't sue me.
Okay. So, um.
Um.
The question is basically that the sequence x_{n} converges to the point x, and it also converges to the point x'. We're meant to discover that these two points are actually the same point. A sequence can't converge to two different points at the same time.
PAUSE FOR A MOMENT AND TRY IT YOURSELF. YOU'LL FEEL VERY SMART.
Let's look at the definition of convergence again.
Convergence: for the sequence x_{n} over the set â€¦
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So, the title sums up everything. When I heard ThisWriter declare me a shapes dude, this sudden wave of impostor syndrome came over me. How could I possibly be a shapes dude? I donâ€™t even know what shapes are! Iâ€™m just some guy who literally thinks that tesseracts are cubes that turn themselves inside out.
The only way to fix this is to, of course, learn all of topology. My estimates suggest that, because topology is easy, it should take me about five minutes. Alright, letâ€™s hop right in. I have some lecture notes right here, so I should just be able to get cracking.
So, chapter 1, section 1. Easy.
Um.
Alright. Back to basics. What is a metric space? I donâ€™t really know. I imagine that itâ€™s a space, and the space presumably has a metric of somâ€¦
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The problems with judging a civilisation by its energy consumption have been laid out well by ThisWriter. The main issue is that Universes, and probably higher verses, are probably infinite; a flat or hyperbolic spacetime (as well as some of the other five Thurston spacetimes that I havenâ€™t brought up) would be infinitely large, and therefore contain an infinite amount of energy.
This isnâ€™t directly an issue, but it leads to two unfortunate consequences:
 There are no friendly qualitative descriptors of Kardashev levels. 1 is Planet, 2 is Star, 3 is Galaxy, but 4 and above donâ€™t mean anything special.
 The qualitative descriptors explode. Universe is Type Infinity. Multiverse is also Type Infinity, but maybe a bigger infinity.
This may be recovâ€¦
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The most natural way to assign infinite numbers to verse levels is of course to use the surreal numbers. The surreal numbers are constructed by having a set of surreals on either side of a bracket. In basic terms, the surreal number obtained as a result is the simplest surreal between those two. As some examples, < x
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The other way to do mechanics, the part that doesn't involve forces and momentum and stuff, is Lagrangians.
Lagrangian mechanics is absolutely crazy, but you'll love it.
The first step is to define your system in terms of a collection of coordinates, and their time derivates. These coordinates depend on whatever's convenient for the system. A particle moving freely in 3D space will typically have these coordinates be x, y, and z. A bob swinging on a pendulum might have one coordinate, Î¸.
The second step is to calculate the kinetic and potential energies T and U of the system when it is at a certain coordinate. The Lagrangian of the system L is then just T  U.
The third step is to find the motion of the system. This is described by the EulerLâ€¦
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