All Questions
Tagged with sequences-and-series cauchy-sequences
921
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Sign permanence of locally Lipschitz functions calculated on a sequence
Suppose I have a sequence $a_k(m)>0$ with $m\in\mathbb{N}$ such that, given $k\in\mathbb{N}$ and $p\geq 1$, I can show that $$|a_{k+p}(m)-a_k(m) |\leq \frac{m^2}{k^2}$$ with $a_{k+p}<a_{k}$. I ...
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Recursive Sequence convergence
Let $(b_n)_n$ be a nonnegative sequence and such that $$\lim_{n\rightarrow\infty}b_{n+1}^2-b_n=a>0$$
Show that $(b_n)_n$ is convergent
My approach is write $a_n=b^2_{n+1}-b_{n}$ and there exists an ...
- 1,833
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13
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Cauchy product of trapezoidal sequences is also trapezoidal
Let $(a_0,a_1,\dots,a_n)$ be a sequence of positive integers such that $a_i=a_{n-i}$ for all $1\leq i\leq n$, and let $m=\lfloor \frac{n}{2}\rfloor$. We say the sequence satisfies the trapezoidal ...
- 309
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2
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Rationals are incomplete and naturals are complete
Why are rationals incomplete and natural complete? I have been going through Analysis textbooks but I don't totally get the reason why.
So, naturals are complete because you can divide them into two ...
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If a random variable converges to zero in probability what can we say about its almost sure boundedness?
First let me start with definitions that I will be using in the question.
A sequence of random variables $X_n(\omega)$ converges to zero in probability if for any $\epsilon>0$, and any $\delta>...
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If $f$ is differentiable at $a$, does it imply that $\lim\limits_{x,y\to a\atop x\ne y} \frac{f(x) - f(y)}{x-y} =f^{\prime}(a)$? [duplicate]
Let $a \in \mathbb{R}$, $I \subseteq \mathbb{R}$ be a neighborhood of $a$, $f: I \rightarrow \mathbb{R}$ a function which is differentiable at $a$.
Want (either/or) :
A function $f$ for which there ...
- 492
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39
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Unable to come up with correct bounds for showing sequence convergence. Analysis I, Terence Tao, Theorem 6.1.19, part (g)
I was trying to prove the following (part (g) of the Theorem):
Theorem 6.1.19 (Limit Laws). Let $(a_n)_{n=m}^{\infty}$ and $(b_n)_{n=m}^{\infty}$ be convergent sequences of real numbers, and
let $x,y$...
- 1,454
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Prove that $(X, d)$ is a complete metric space where $X$ is the set of all real sequences and $d: X \times X \to \mathbb{R}$ defined by...
I am given a metric space $(X, d)$ where $X$ is the set of all real sequences and $d: X \times X \to \mathbb{R}$ the metric defined by
$$
d(x, y) = \begin{cases}(\sup\{n \in \mathbb{N}: x_k = y_k \...
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Understanding a proof that a uniformly Cauchy sequence of continuous functions $\{f_n\}_n$ converges uniformly to a limit function $f$
Suppose $\{f_n\}_n$ is a uniformly Cauchy sequence of continuous function $f_n:\mathbb{R}\to S$ for a complete metric space $S$. I am trying to understand the "standard" proof that the ...
- 1,221
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Proposition 5.4.9. Analysis I - Terence Tao.
Proposition 5.4.9 (The non-negative reals are closed). Let $a_1, a_2, a_3, \ldots $ be a Cauchy sequence of non-negative rational numbers. Then $\text{LIM}_{n \to \infty}a_n$ is a non-negative real ...
- 1,454
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1
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Cauchy sequences over $\mathbb{Q}$: A basic question
Define an equivalence relation on Cauchy sequences in $\mathbb{Q}$ by $(x_n)\equiv (y_n)$ if $\lim (x_n-y_n)=0$.
The question I am trying to prove is that
If $(x_n)$ and $(y_n)$ are Cauchy sequences ...
- 3,047
2
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1
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78
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Given a sequence $a_n$ s.t. forall $\varepsilon>0$ exists $m,n_{0} \in \mathbb{N}$ s.t. $\forall n \geq n_0$ $|a_n-a_m|<\varepsilon$
So I have that question:
Suppose a sequence $a_n$ satisfies that forall $\varepsilon>0$ exists $m,n_{0} \in \mathbb{N}$ s.t. $\forall n \geq n_0$ $|a_n-a_m|<\varepsilon$ determine if that ...
- 115
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Not decreasing sequence that converges to 0: sufficient condition for the sequence of indexes for which the sequence decreases to be bounded.
Consider a strictly positive sequence $\{u_n\}$ of real numbers (i.e. $\forall n, u_n >0$). Suppose that $\displaystyle \lim_{n \to \infty} u_n = 0$ and that $\{u_n\}$ is not decreasing.
Consider ...
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1
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46
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Trying to construct a subsequence
Let $(x_n)$ be a real valued sequence. I want to construct a subsequence $(y_n) $ of $(x_n) $ such that no two consecutive terms of the sequence $(
y_n) $ are same and also if $(y_n) $ is Cauchy then $...
- 3,664
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How is contradiction with definition of product of two sequences possible?
I read about Infinitesimal Differential Geometry in this source, page 238-239 and I considered one example while I was reading. I was confused because in this example we have contradiction with ...
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