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 1d awarded Necromancer Jan11 awarded calculus Sep30 awarded Explainer Sep24 awarded Autobiographer Sep11 comment Any open subset of $\Bbb R$ is a at most countable union of disjoint open intervals. [Collecting Proofs] @user10444: It is countable yes! If you agree that $\{(\alpha_x,\beta_x):x\in U\}$ is a disjoint family of open intervals then you can see it by choosing $r_x\in \mathbb Q\cap (\alpha_x,\beta_x)$ for all $x\in U$. Then $\{r_x:x\in U\}$ is countable right? Note that the intervals $\{(\alpha_x,\beta_x):x\in U\}$ are not distinct! Sep11 awarded Informed Sep7 awarded Yearling Aug11 answered Find polynomial whose root is sum of roots of other polynomials Aug11 revised Generalized mean value theorem deleted 5 characters in body Apr16 revised How to calculate square matrix to power n? edited body Apr13 comment Why does $\gcd(a, b) = \min\lbrace ma + nb : m, n\in\mathbb{Z}\text{ and }ma+nb>0\rbrace$? @DwayneE.Pouiller See my edited answer. Apr13 revised Why does $\gcd(a, b) = \min\lbrace ma + nb : m, n\in\mathbb{Z}\text{ and }ma+nb>0\rbrace$? added 119 characters in body Jan26 revised Generalized mean value theorem added 5 characters in body Jan11 answered Prove that if $d$ divides $n$ then $φ(d)$ divide $φ(n)$ for $φ$ denotes Euler’s φ-function. Jan8 comment How find this sequence $\{x_{n}\}$ such $\lim_{n\to \infty}x_{n}\left(1-\frac{n(1-na_{n})}{\ln{n}}\right)=1$ What about $x_{n}=\frac{1}{\left(1-\dfrac{n(1-na_{n})}{\ln{n}}\right)}$? Dec24 comment Provide a proof using the rational roots theorem I suppose Matina Manos you mean $c$ is a positive integer! Dec24 revised Provide a proof using the rational roots theorem added 7 characters in body; edited title Dec24 comment Number of integral roots of a polynomial @posthumus: Since $(x-d)\mid p(x)$ it follows that $(a-d)\mid p(a)=-1$. Therefore $(a-d)\mid -1$ and similarly $(b-d)\mid -1, (c-d)\mid -1$. Also $a-d, b-d, c-d$ are $3$ different integers. Now how many factors does $-1$ have? Dec19 revised Find $\lim_{x\to\infty} \frac{(x+1)^2(2x-4)^3}{(2x-1)^4}$ deleted 6 characters in body Dec16 reviewed Leave Open Optimization using Quadratic form