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This is a very elementary topic but I thought it might be worth giving it a try here, I would be very interested in any comments - I originally posted it to Maths SE. Euler's Rotation Theorem, proved …

Two geometric proofs are given below. Both proofs start off as in Euler's proof, by considering the image $C_{2}$ of a great circle $C_{1}$ under the motion. In proof (1) this is used to construct a n …

Is it possible to solve the following two problems on intervals using elementary methods, without using the outer measure ? Problem 1 If $(I_n)$ is a disjoint sequence of subintervals of interval $I$ …

Further to Nik Weaver's answer, which proves the case of $I$ bounded in Problem 1, the unbounded case is proved below - it follows as a corollary of the bounded case. I will update this answer later i …

The following proof begins by considering some simple cases, and then extends to the general case. The general case taken alone gives a short proof of the theorem, but the simple cases provide a motiv …

What are some alternative methods of proof for the necessity direction of the above theorem, ie $n$ an even perfect number $\Rightarrow n$ is of form $2^{a-1} (2^a - 1)$ where $2^a - 1$ is a Mersenne …