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(Un-)Countable union of open sets - Mathematics Stack Exchange 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
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The sequence of integers - Mathematics Stack Exchange 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
Prove that the sequence (1+1 n)^n is convergent [duplicate] 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 the process of
functional analysis - Where can I find the paper Un théorème de . . . 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)
Divergence of series sin (1 n) - Mathematics Stack Exchange Since −1 ≤ sin(1 n) ≤ 1 1 ≤ sin (1 n) ≤ 1 and limn→∞ −1 lim n → ∞ 1 ≠ ≠ limn→∞ 1 lim n → ∞ 1 can I use the nth-term test to prove that the series will diverge? I've only seen the problem done using the limit comparison test and am not sure if I can use the nth-term test
How to parametrize a triangle? - Mathematics Stack Exchange How do I parametrize a triangle with vertices A(1, 1) A (1, 1), B(2, 2) B (2, 2) and C(1, 3) C (1, 3)? I have tried working with the equations of the lines that form it but am not completely sure how to link them together to come up with a parametrization of the form r(t) r (t)
modular arithmetic - Prove that that $U (n)$ is an abelian group . . . 1 Let a ∈ Un a ∈ U n then we have to show that there exists b ∈ Un b ∈ U n such that a b a b mod n = 1 n = 1 Let us suppose o(a) = p ap = e o (a) = p a p = e Now if b b is inverse of a a then a b a b mod n = 1 n = 1 holds i e a b = x(n) + 1 a b = x (n) + 1 for some x x (By division algorithm) Now multiply ap−1 a p 1