When N=2 we have for example \varepsilon_{12}=+1. Now, you didn't define \varepsilon^{12} for us. Also, is there some Einstein summation convention in effect here?
You could, instead, do this:
\frac{A}{x-1} + \frac{Bx+C}{(x-2)^2}.
The point is that the numerator can be anything with degree less than the degree of the denominator.
When you say p-adic analysis, p is a prime, so p=0 is not used. |0|_p = 0. Sometimes the usual absolute value |x| is called the \infty-adic absolute value, and \infty is listed among the "primes". The p-adic absolute value is defined for the p-adic numbers, not the real numbers. Except the...
The covectors constitute the dual space of the vectors. This tells mathematicians how to treat them when you change variables. Physicists have these weird formulas to use for that... Why? You'll have to ask a physicist.
Wait: there's a bijection between the integers and the even integers, even though the big one has TWO copies (evens and odds) of the small one. Cardinality does these strange things.
Nope. You have not proved the length of the circle (your limiting curve) is 4. You have proved, instead, that the length of the limit is not equal to the limit of the lengths. No need to use a circle to prove this: For example, you can use stairsteps converging to a sloping line segment.