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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.
Apr
13
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
Jan
26
revised Generalized mean value theorem
added 5 characters in body
Jan
11
answered Prove that if $d$ divides $n$ then $φ(d)$ divide $φ(n)$ for $φ$ denotes Euler’s φ-function.
Jan
8
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 $x_{n}=\frac{1}{\left(1-\dfrac{n(1-na_{n})}{\ln{n}}\right)}$?
Dec
24
comment Provide a proof using the rational roots theorem
I suppose Matina Manos you mean $c$ is a positive integer!