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A remark: regardless of whether it is true that an infinite union or intersection of open sets is open, when you have a property that holds for every finite collection of sets (in this case, the union or intersection of any finite collection of open sets is open) the validity of the property for an infinite collection doesn't follow from that. In other words, induction helps you prove a ...

Nov 11, 2018 · The Integration by Parts formula may be stated as: $$\\int uv' = uv - \\int u'v.$$ I wonder if anyone has a clever mnemonic for the above formula. What I often do is to derive it from the Product R...

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May 9, 2016 · Prove that the sequence $\\{1, 11, 111, 1111, .\\ldots\\}$ will contain two numbers whose difference is a multiple of $2017$. I have been computing some of the immediate multiples of $2017$ to see how

Mar 27, 2019 · It is hard to avoid "the concept of calculus" since limits and convergent sequences are a part of that concept. On the other hand, it would help to specify what tools you're happy with using, since this result is used in developing some of them. (For example, if you define ex = limn→∞(1 + x/n)n e x = lim n → ∞ (1 + x / n) n, then clearly we should not be using ex e x in …

Aug 3, 2023 · It might be using ring theory in a non-essential way, but it is conceptually simpler because the endomorphisms are easier to describe than the automorphisms, and since the invertible elements of Zn Z n are by definition Un U n, we obtain the result without having to understand what Un U n actually looks like.

Nov 12, 2015 · J. P. Aubin, Un théorème de compacité, C.R. Acad. Sc. Paris, 256 (1963), pp. 5042–5044. It seems this paper is the origin of the "famous" Aubin–Lions lemma. This lemma is proved, for example, here and here, but I'd like to read the original work of Aubin. However, all I got is only a brief review (from MathSciNet).

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Jan 5, 2016 · and what you'd really like is for an isomorphism U(n) ≅ SU(n) × U(1) U (n) ≅ S U (n) × U (1) to respect the structure of this short exact sequence. (If there were some random isomorphism that didn't have this property that would be less interesting.) For starters, this requires that det: U(n) → U(1) d e t: U (n) → U (1) have a section, or equivalently that the short …