Questions tagged [cauchy-sequences]
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For questions relating to the properties of Cauchy sequences.
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Cauchy net equivalent condition
Let $E$ be a topologic vector space. $x_i$ with $i\in I$ is a Cauchy net iff $\lim_j \ x_{\phi(j)}-x_{\psi(j)}=0$ for any $(\phi,\psi)$ pair of cofinal and increasing maps from $J$ to $I$. I proved ... general-topology analysis cauchy-sequences topological-vector-spaces nets- 31
Baby Rudin theorem 11.42
There are the definitions which we need for the proof of the theorem : There is the theorem: If ${f_n}$ is a Cauchy sequence in $\mathscr L^2(\mu)$ , then there exists a function $f$ $\in$ $\mathscr ... integration functional-analysis measure-theory multivariable-calculus cauchy-sequences- 427
Is $(f_k)$ a Cauchy sequence on $C^0_∞[−1, 1]$ under the Supremum Norm?
I’m trying to show that the following sequence of functions is or is not a Cauchy sequence on $C^0[-1,1]$ space under $\|.\|_{\infty}$ but I cannot obtain something concrete. Consider the sequence of ... functional-analysis cauchy-sequences- 11
Equivalent condition for Cauchy Sequences
I am trying to prove the following : Let $E$ be a topologic vector space. $(x_n)\subset E$ is a Cauchy sequence in $E \ $ iff $\ \lim_{k\to\infty}(x_{m_k}-x_{n_k})=0$ for any $n_k, m_k$ pair of ... general-topology analysis cauchy-sequences topological-vector-spaces- 63
Prove a sequence is Cauchy's by Induction [closed]
If $x_1=1,x_2=2$, then the sequence $(x_n)=\frac{1}{2}(x_{n-2}+x_{n-1})$ for $n>2$. We want to prove by Induction that $|x_n-x_{n+1}|=\frac{1}{2^{n-1}}$ for $n\in \Bbb N$ (Cauchy sequence).. I ... real-analysis induction cauchy-sequences- 67
Decimal Cauchy sequence
Given a real number $x\in\mathbb{R}$ in decimal form $$ x = C_0. C_1 \dotsc, \quad C_0 \in \mathbb{Z}, \quad C_i = 0, \dotsc, 9, \quad i > 0 $$ we may define a sequence of decimals $$ q_n = C_0. ... calculus cauchy-sequences decimal-expansion- 11
Intuitionistic Disproof of Intermediate Value Theorem
For uni I'm studying intuitionism, and I came across the following disproof for the IVT: The thing I'm trying to understand is why this disproof is not valid in classical mathematics. In my research ... real-analysis cauchy-sequences infinity constructive-mathematics- 129
Continuous linear functionals on $\{ c\in\mathbb{C}^{\mathbb{N}}: \forall n\in\mathbb{N}, \sup_k |c_k|(1+k)^n<\infty\}$
Consider the space $S=\{c=(c_k)_{k\in\mathbb{N}}\in \mathbb{C}^{\mathbb{N}}: \forall n\in\mathbb{N},\sup_{k\in\mathbb{N}}|c_k|(1+k)^n<\infty\}$ endowed with the locally convex topology generated by ... functional-analysis cauchy-sequences dual-spaces- 2,246
Understanding how to use a Cauchy sequence to construct an arbitary number.
I have been a bit confused with how Cauchy sequences work. How might I construct a Cauchy Sequence to construct something non-obvious, like $\sqrt{2.5}$ I feel like there's something going over my ... sequences-and-series cauchy-sequences- 101
Why is this infinite series _intuitionistically_ Cauchy?
I'm currently writing a short paper on Intuitionism for uni. The subject of this paper is the decay of the intermediate value theorem under intuitionism. I have found a proof for this but I have a ... cauchy-sequences intuitionistic-logic goldbachs-conjecture- 129
Prove uniform convergence of bounded sequence
Supose we have a sequence of real continuous functions on an interval $[0,b] \in \mathbb{R}$ $(f_n)_{n=0}^{\infty}$ (hence uniformly continuous). Moreover the sequence has the following properties: ... real-analysis sequences-and-series uniform-convergence cauchy-sequences- 608
Prove that every cauchy-sequence converges in $\mathbb{R^n}$ by concluding from the nested intervals theorem
I am preparing for my exam by doing some exercises and need help with the following task: Prove that every cauchy-sequence is convergent in $\mathbb{R^n}$ by using the nested intervals theorem. Hint:... real-analysis general-topology cauchy-sequences- 292
Show that $\{x\in\mathbb{C}^{\mathbb{N}}: \lim_{n\to\infty} \frac{x_n}{a_n}=0\}$ is a Banach space
Consider $a:=\{a_n\}_{n=0}^{\infty}\in\mathbb{R}^{\mathbb{N}}_{+}$ a decreasing sequence with $a_n\to 0$. Consider the following space of sequences $$ X_a:=\left\{x:=\{x_n\}_{n=0}^{\infty}\in\mathbb{C}... banach-spaces cauchy-sequences complete-spaces- 2,246
How do you come to the conclusion $0<1/(m+1)!+\dots+1/n!<1/(2^n)$ from $2^k<k!$ if $k\geq 4$?
I'm working out of the Bartle and Sherbert Introduction to Real Analysis book and I'm looking at the partial solution to proving the sequence $(1+1/2!+\dots+1/n!)$ is Cauchy (Exercise 3.5.2(b)). Their ... real-analysis sequences-and-series cauchy-sequences- 661
Show that if $0 < \frac{1}{j},\frac{1}{k} \leq \frac{1}{N} \leq 1$, then $|\frac{1}{j}-\frac{1}{k}| \leq \frac{1}{N}$. ($j,k,N \in \mathbb{N}^+$))
I am working through Tao's analysis I book and I am trying to prove that the sequence $a_n = \frac{1}{n}$ is a Cauchy sequence (Proposition 5.1.11). I understand the proof, but I am having trouble ... analysis absolute-value cauchy-sequences- 33
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