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In this paper we give an elementary proof for Bertrand's postulate also known as Bertrand-Chebyshev theorem.
We provide yet another proof of the existence of calibrated forecasters; it has two merits. First, it is valid for an arbitrary finite number of outcomes. Second, it is short and simple and it follows from a direct application of…
We prove Union-Closed sets conjecture.
We present a method to prove the decidability of provability in several well-known inference systems. This method generalizes both cut-elimination and the construction of an automaton recognizing the provable propositions.
This article offers a gentle introduction to the axiom of choice. We introduce the axiom, discuss some common objections to it, and present three kinds of reasons to accept it. Although the exposition is aimed at non-experts in set theory,…
A very short proof of the Fej\'er-Riesz lemma is presented in the matrix case
We give a proof of the Marker-Steinhorn Theorem which fills a gap in previous proofs of the result.
We prove a local version of the Mazur-Ulam theorem.
We settle in the affirmative the Graham-Sloane conjecture.
We consider several examples of probabilistic existence proofs using compressibility arguments, including some results that involve Lov\'asz local lemma.
We study a well-known technique of using absoluteness for giving choice-free proofs to some statements which are known to be provable with the axiom of choice. The idea is to reduce the problem to an inner model where the axiom of choice…
These informal notes, not intended for publication, provide an approach to the Borsuk--Ulam theorem via Stokes' theorem, in a similar spirit to Lima's proof of the Brouwer fixed point theorem. They are intended to be accessible to anyone…
Based on various strategies and a new general doubling operator, we obtain several simple proofs of the celebrated Sharkovsky's cycle coexistence theorem. A simple non-directed graph proof which is especially suitable for a calculus course…
We give an elementary probabilistic proof of a binomial identity. The proof is obtained by computing the probability of a certain event in two different ways, yielding two different expressions for the same quantity.
We present a new proof of the celebrated quadratic reciprocity law. Our proof is based on group theory.
We give a simple proof of the increasing strengthening of Arhangel'skii's Theorem. Our proof naturally leads to a refinement of this result of Juh\'asz.
Here we give a short survey of our new results. References to the complete proofs can be found in the text of this article and in the litterature.
This note presents an elementary and direct proof for the convexity of the Choquet integral when the corresponding set function is submodular.
Oftentimes, Stokes' theorem is derived by using, more or less explicitly, the invariance of the curl of the vector field with respect to translations and rotations. However, this invariance -- which is oftentimes described as the curl being…
We provide elementary proof of several congruences involving single sum and multisums of binomial coefficients.