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 Sep 14 awarded Good Answer Sep 7 awarded Yearling Aug 27 awarded Nice Answer Aug 13 awarded Nice Answer Jul 12 awarded Necromancer May 22 comment Is a Cauchy sequence - preserving (continuous) function is (uniformly) continuous? No the contrapositive is "If $f$ is not continuous then it is not true that 'sends Cauchy sequences to Cauchy sequences'" or equivalently "If $f$ is not continuous then there is at least one Cauchy sequence $(a_n)_{n\in\mathbb N}$ such that $(f(a_n))_{n\in\mathbb N}$ is not a Cauchy sequence" May 22 comment Is a Cauchy sequence - preserving (continuous) function is (uniformly) continuous? For which one do you mean specific. The $(a_n)_{n\in\mathbb N}$ or the $(b_n)_{n\in\mathbb N}$? If you mean the $(b_n)_{n\in\mathbb N}$ then the answer is No is not true for all the Cauchy sequences. For example if you take a constant sequence then is not true. The $(a_n)_{n\in\mathbb N}$ is not specific is general. May 22 comment Is a Cauchy sequence - preserving (continuous) function is (uniformly) continuous? @VikrantDesai The sequence $(f(b_n))_{n\in\mathbb N}$ is not Cauchy because $\rho(f(b_{2n}),f(b_{2n-1}))=\rho(f(a_{2n}),f(x))>\epsilon$ for all $n\in\mathbb N$. May 22 comment Is a Cauchy sequence - preserving (continuous) function is (uniformly) continuous? @VikrantDesai This is true only when the metric space $X$ is complete. Apr 25 awarded Necromancer Jan 11 awarded calculus Sep 30 awarded Explainer Sep 24 awarded Autobiographer Sep 11 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! Sep 11 awarded Informed Sep 7 awarded Yearling Aug 11 answered Find polynomial whose root is sum of roots of other polynomials Aug 11 revised Generalized mean value theorem deleted 5 characters in body Apr 16 revised How to calculate square matrix to power n? edited body Apr 13 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.