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sub-sequence

sub-sequence2 (ˈsʌbsiːkwəns) [sub- 7 a, e.] A sequence contained in or forming part of another sequence; spec. in Math.1908 [see oscillatory a. 3]. 1958 R. C. Moore Introd. Hist. Geol. (ed. 2) iv. 80 The second division of the Huronian Sequence, named the Cobalt Sub-sequence, has an aggregate thickn... Oxford English Dictionary

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Showing a sub-sequence ($r_{n_{k}}$) converges to $x$ There exists a bijection that $f : N → Q, x \in R$ *$r_{n}$:=$f(n)$ I am asked to show that there exists a sub-sequence ($r_{n_{k}}$) of ($r_{n}$) so that the l...

Let $n_1 < n_2 < n_3$ ,then the limit of $i$ will converges to x. By the density of $Q$ , there will be a $n_1$ that $x−1$ < $n_1$ < $x+1$ Therefore, $x− 1$ < $r_n$ < $x+ 1$ Notices that there exists infinitely many rational numbers belong to the interval and hence, it is always possible.

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Does every convergent sequence have a sub-sequence whose terms comes closer than any positive sequence? Let $(x_n)$ be convergent sequence of real numbers and $(y_n)$ be any sequence of positive real numbers , then is...

By definition of convergence (actually, by using the fact that any convergent sequence is a cauchy sequence), $\forall \epsilon > 0, \exists N\in \mathbb{N}: \forall n, m > N, |x_n - x_m| N_1, |x_n - x_m| N_2, N_2 \geq N_1, |x_n - x_m| < y_2$$ You'll get your subsequence $(x_{r_n})$ in question.

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Existence of a sub-sequence of non-converge sequence that $|x_{p_n}-L|>\epsilon$ > Let $\\{x_n\\}$ be a sequence that does not converge and let L be a real number. Prove that there exist $\epsilon >0$ and a sub-sequen...

Since $(x_n)$ does not converge to $L$ there is $\epsilon >0$ such that $|x_n-L|> \epsilon$ for infinitely many n. Now, its your turn.

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Does every bounded sequence have a Cauchy sub-sequence? In an answer to an earlier question it was explained why a bounded sequence is not guaranteed to be a Cauchy sequence. But does every bounded sequence have a Ca...

No, it is not true in general that every bounded sequence has a Cauchy subsequence. Define a metric $d$ on $\Bbb R$ by $d(x,y)=\min\\{|x-y|,1\\}$, and consider the sequence $\sigma=\langle n:n\in\Bbb N\rangle$. Clearly $d(m,n)=1$ whenever $m,n\in\Bbb N$ and $m\ne n$, so $\sigma$ has no Cauchy subseq...

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Prove the convergence of a sequence based on sub-sequence Suppose that every convergent sub-sequence of $a_n$ converges to zero. Prove that $a_n$ converges to zero. I tried to prove this statement using the definitio...

First of all, $(a_{n})$ must be bounded, if not, one can find some $|a_{n_{k}}|$ such that $|a_{n_{k}}|\rightarrow\infty$, if your convergence includes the sort of infinity, then this violates the assumption. Now let a subsequence $(a_{n_{k}})$ such that $\lim_{k\rightarrow\infty}|a_{n_{k}}|=\limsup...

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If a sequence has a convergent sub-sequence, then the sequence converges(fake proof) Let $(X,d)$ be a metric space and $x_n$ be a sequence such that it has a convergent sub-sequence $x_{n_k} \to x $, then $x_n \to x$ ...

The contrapositive is wrong. To say $x_n \not\rightarrow x$ is to say \begin{equation} \exists\epsilon>0 \text{ s.t. } \forall N \in \mathbb{N} \: \: \exists n \geq N \text{ s.t. } d(x_n,x) \geq \epsilon \end{equation} And not \begin{equation} \exists\epsilon>0 \text{ s.t. } \forall N \in \mathbb{N}...

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Bounded sequence in $W^{1,p}(I)$ that has no convergent sub-sequence in $L^{\infty}$ Let $I \subset \mathbb{R}$ be an open interval and $p \geqslant 1$. Is there a bounded sequence in $W^{1,p}(I)$ that has no converge...

For $p=1$, yes. Take $I = (0,1)$ and let $f_n(x) = \begin{cases} nx, & x 1$, no. This is a standard Sobolev embedding theorem, but in this case you can prove it with Arzela-Ascoli. Recall that every function in $W^{1,p}(I)$ has a continuous version, so we work with those. Now use Holder's inequality...

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What is the probability that a sequence of $M$ random letters contains a sub-sequence of $K$ letters? The alphabet is supposed to have $n$ characters (usually $n=26$, though in my case $n=256$ or $256^2$ or $256^4$). ...

If you just need an _upperbound_ , that is much easier. Let $X =$ the number of times the sequence appears. There are $M-K+1$ possible starting possitions, so by linearity of expectation, $E[X] = (M - K + 1) n^{-K} < M n^{-K}.$ Now $E[X] = \sum_{k \ge 0} k P(X = k) = \sum_{k \ge 1} k P(X = k) \ge \s...

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Expected length of the "greedy" increasing sub-sequence? Given a sequence of random unique integers of length $n$, if I select every element that is the largest so far how how many elements should I expect to select? ...

From symmetry considerations, it's easy to see that the $k$th element has exactly a $1/k$ chance of being the largest among the first $k$ terms. By linearity of expectation, the expected number of "record-breakers" in a random ordering is just $\sum_{k=1}^n \frac1k$. This is the $n$th harmonic numbe...

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Prove that $\cos (nx)$ has no sub-sequence converging uniformly Let $f_n(x)=\cos(nx)$ on $\Bbb{R}$. Prove that there is no sub-sequence $f_{n_k}(x)$ converging uniformly in $\Bbb{R}$. I have no clue how to address thi...

Hint: for any $n \ne m$, there is some $x$ such that $|\cos(nx) - \cos(mx) |\ge 1$.

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Two questions: Construction of sequences and Subsequence I have two questions regarding sequences: 1. I was asked to construct a sequence such that for any given $n \in \mathbf{N}$, the sequence has a sub-sequence ...

Now, suppose there is no sub-sequence $x_{n_i}$ such that $ ∀i,x_{n_i}=x_{n_1}$. This means there is no sub-sequence of ${x_n}$ which is constant. So no term is repeated infinitely many times ( to form a sub-sequence).

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How do I prove that if $L$ is sub-sequential limit of $a_{n_k}$ then $L$ is also sub-sequential limit of $a_n$? Could you please give me some hint how to solve this simple question: > Suppose $a_{n_k}$ is sub-sequenc...

Let $(a_n)_{n\in I}$ be a sequence, and let $(a_{n_k})_{n_k\in J}$, for some subset $J\subseteq I$ of indices be a subsequence. $L$ is a subsequential limit of $(a_{n_k})_{n_k\in J}$ if there exists a subsequence $(a_{n_{k_m}})_{n_{k_m}\in K}$ that converges to $L$. By the definition of a subsequenc...

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Understanding compact subsets of metric spaces Please help me understand the following definition: > Let $(X,d)$ be a metric space, a subset $S \in X$ is called _compact_ , if any infinite sequence $\\{x_{n}\\}_{n\in...

The definition can be reformulated as follows - "Let $(X,d)$ be a metric space, a subset $S∈X$ is called compact, if for all infinte sequences $\\{ x_{n}\\}_{n=1}^{∞}\subseteq S$ the following holds: $\\{ x_{n}\\}_{n=1}^{∞}$ has a concentration point and if $\bar{x}$ is a concentration point of $\\{...

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sub-sequence

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sub-sequence

Showing a sub-sequence ($r_{n_{k}}$) converges to $x$ There exists a bijection that $f : N → Q, x \in R$ *$r_{n}$:=$f(n)$ I am asked to show that there exists a sub-sequence ($r_{n_{k}}$) of ($r_{n}$) so that the l...

Does every convergent sequence have a sub-sequence whose terms comes closer than any positive sequence? Let $(x_n)$ be convergent sequence of real numbers and $(y_n)$ be any sequence of positive real numbers , then is...

Existence of a sub-sequence of non-converge sequence that $|x_{p_n}-L|>\epsilon$ > Let $\\{x_n\\}$ be a sequence that does not converge and let L be a real number. Prove that there exist $\epsilon >0$ and a sub-sequen...

Does every bounded sequence have a Cauchy sub-sequence? In an answer to an earlier question it was explained why a bounded sequence is not guaranteed to be a Cauchy sequence. But does every bounded sequence have a Ca...

Prove the convergence of a sequence based on sub-sequence Suppose that every convergent sub-sequence of $a_n$ converges to zero. Prove that $a_n$ converges to zero. I tried to prove this statement using the definitio...

If a sequence has a convergent sub-sequence, then the sequence converges(fake proof) Let $(X,d)$ be a metric space and $x_n$ be a sequence such that it has a convergent sub-sequence $x_{n_k} \to x $, then $x_n \to x$ ...

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Bounded sequence in $W^{1,p}(I)$ that has no convergent sub-sequence in $L^{\infty}$ Let $I \subset \mathbb{R}$ be an open interval and $p \geqslant 1$. Is there a bounded sequence in $W^{1,p}(I)$ that has no converge...

What is the probability that a sequence of $M$ random letters contains a sub-sequence of $K$ letters? The alphabet is supposed to have $n$ characters (usually $n=26$, though in my case $n=256$ or $256^2$ or $256^4$). ...

Expected length of the "greedy" increasing sub-sequence? Given a sequence of random unique integers of length $n$, if I select every element that is the largest so far how how many elements should I expect to select? ...

Prove that $\cos (nx)$ has no sub-sequence converging uniformly Let $f_n(x)=\cos(nx)$ on $\Bbb{R}$. Prove that there is no sub-sequence $f_{n_k}(x)$ converging uniformly in $\Bbb{R}$. I have no clue how to address thi...

Two questions: Construction of sequences and Subsequence I have two questions regarding sequences: 1. I was asked to construct a sequence such that for any given $n \in \mathbf{N}$, the sequence has a sub-sequence ...

How do I prove that if $L$ is sub-sequential limit of $a_{n_k}$ then $L$ is also sub-sequential limit of $a_n$? Could you please give me some hint how to solve this simple question: > Suppose $a_{n_k}$ is sub-sequenc...

Understanding compact subsets of metric spaces Please help me understand the following definition: > Let $(X,d)$ be a metric space, a subset $S \in X$ is called _compact_ , if any infinite sequence $\\{x_{n}\\}_{n\in...