What is Subsequence: Definition and 60 Discussions
In mathematics, a subsequence is a sequence that can be derived from another sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence
⟨
A
,
B
,
D
⟩
{\displaystyle \langle A,B,D\rangle }
is a subsequence of
⟨
A
,
B
,
C
,
D
,
E
,
F
⟩
{\displaystyle \langle A,B,C,D,E,F\rangle }
obtained after removal of elements
C
{\displaystyle C}
,
E
{\displaystyle E}
, and
F
{\displaystyle F}
. The relation of one sequence being the subsequence of another is a preorder.
Subsequences can contain consecutive elements which were not consecutive in the original sequence. A subsequence which consists of a consecutive run of elements from the original sequence, such as
⟨
B
,
C
,
D
⟩
{\displaystyle \langle B,C,D\rangle }
from
⟨
A
,
B
,
C
,
D
,
E
,
F
⟩
{\displaystyle \langle A,B,C,D,E,F\rangle }
, is a substring. The substring is a refinement of the subsequence.
The list of all subsequences for the word "apple" would be "a", "ap", "al", "ae", "app", "apl", "ape", "ale", "appl", "appe", "aple", "apple", "p", "pp", "pl", "pe", "ppl", "ppe", "ple", "pple", "l", "le", "e", "" (empty string).
Let's us look at the first implication (I will post the reverse implication once this proof has been verified). We have to prove if there is a subsequence of ##(s_n)## converging to ##t##, then there are infinitely many elements of ##(s_n)## lying within ##\epsilon## of ##t##, for any...
Let's say we're given a sequence ##(s_n)## such that ##\lim s_n = s##. We have to prove that all subsequences of it converges to the same limit ##s##. Here is the standard proof:
Given ##\epsilon \gt 0## there exists an ##N## such that
$$
k \gt N \implies |s_k - s| \lt \varepsilon$$
Consider...
Hello i have problems with this exersice
Let $$\{X_{\alpha}\}_{\alpha \in I}$$ a collection of topological spaces and $$X=\prod_{\alpha \in I}X_{\alpha}$$ the product space. Let $$p_{\alpha}:X\rightarrow X_{\alpha}$$, $$\alpha\in I$$, be the canonical projections
a)Prove that a sequence...
Problem: If sequence ## (a_n) ## has ##10-10## as partial limits and in addition ##\forall n \in \mathbb{N}.|a_{n+1} − a_{n} |≤ \frac{1}{n} ##, then 0 is a partial limit of ## (a_n) ##.
Proof : Suppose that ## 0 ## isn't a partial limit of ## (a_n) ##. Then there exists ## \epsilon_0 > 0 ## and...
Summary:: x
Let ## \{ a_{n} \} ## be a sequence.
Prove: If for all ## N \in { \bf{N} } ## there exists ## n> N ## such that ## a_{n} \leq L ## , then there exists a subsequence ## \{ a_{n_{k}} \} ## such that ##
a_{n_{k}} \leq L ##
My attempt:
Suppose that for all ## N \in {\bf{N}} ##...
Hello! (Wave)
Let $(a_n), (b_n), (c_n)$ sequences such that $(a_n), (c_n)$ are bounded and $a_n \leq b_n \leq c_n$ for each $n=1,2, \dots$ I want to show that $(b_n)$ has a convergent subsequence.
I have thought the following:
Since $(a_n), (c_n)$ are bounded, $\exists m_1, m_2 \in...
Homework Statement
Let ##\{a_n\}## be a bounded sequence such that every convergent subsequence has limit ##L##. Prove that ##\lim_{n\to\infty}a_n = L##.
Homework EquationsThe Attempt at a Solution
I'm not really understanding this problem. Isn't ##\{a_n\}## a subsequence of itself? So isn't...
Homework Statement
Prove that every convergent sequence has a monotone subsequence.
Homework EquationsThe Attempt at a Solution
Define ##L## to be the limit of ##(a_n)##. Then every ##\epsilon##-ball about L contains infinitely many points. Note that ##(L, \infty)## or ##(-\infty, L)## (or...
##X_i## is an independent and identically distributed random variable drawn from a non-negative discrete distribution with known mean ##0 < \mu < 1## and finite variance. No probability is assigned to ##\infty##.
Now, given ##1<M##, a sequence ##\{X_i\}## for ##i\in1...n## is said to meet...
What I have is
/// <summary>
/// Provides a solution to the Common Child string problem
/// https://www.hackerrank.com/challenges/common-child/problem
/// </summary>
public static class CommonChild
{
public static int Solve(string first, string second)...
Homework Statement
"Let ##\{a_n\}_{n=1}^\infty## be a bounded, non-monotonic sequence of real numbers. Prove that it contains a convergent subsequence."
Homework Equations
Monotone: "A sequence ##\{\alpha_n\}_{n=1}^\infty## is monotone if it is increasing or decreasing. In other words, if a...
Homework Statement
For some background, this is an advanced calculus 1 course. This was an assignment from a quiz back early in the semester. Any hints or a similar problem to guide me through this is greatly appreciated! Here is the problem:
Find a convergent subsequence of the sequence...
Homework Statement
Consider the space ##([0, 1], d_1)## where ##d_1(x, y) = |x-y|##. Show that there exists a sequence ##(x_n)## in ##X## such that for every ##x \epsilon [0, 1]## there exists a subsequence ##(x_{n_k})## such that ##\lim{k\to\infty}\space x_{n_k} = x##.
Homework Equations
N/A...
Let $\left\{{x}_{n}\right\}$ be a sequence...İf $\left\{{x}_{2n}\right\}$ is caucy sequence, can we say that $\left\{{x}_{n}\right\}$ is cauchy sequence ?
Homework Statement
1. Consider the sequence $$\frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5},\frac{1}{6}, \ldots$$ For which values ##z \in \mathbb{R}## is there a subsequence converging to ##z##?
2. Prove that...
Hello,
Solving last exam and stuck in this exercise
Homework Statement
Consider an increasing sequence {xn} . We suppose ∃ x∈ℝ and {xnk} a sebsequence of {xn} and xnk→x
a/ Show that for any n∈ℕ , ∃ k∈ℕ as n≤nk
b/ Show that xn→x
Homework Equations
3. The Attempt at a Solution [/B]
For b/ it...
I'm not sure if this is true or not. but from what I can gather, If the set of Natural numbers (divergent sequence) {1, 2, 3, 4, 5,...} is broken up to say {1}, is this a subsequence that converges and therefore this statement is true?
Hello! (Wave)
Let $X=<x_1,x_2, \dots , x_m>$ and $Y=<y_1,y_2, \dots, y_n>$ be sequences and let $Z=<z_1, z_2, \dots, z_k>$ a greatest common subsequence(GCS) of $X$ and $Y$.Then:
If $x_m=y_n$,then $z_k=x_m=y_n$ and $Z_{k-1}$ is a GCS of $X_{m-1}$ and $Y_{n-1}$.
If $x_m \neq y_n$ and $z_k...
Hi ! :)
Let x_{n} a sequence of positive numbers.How could I show that it has a decreasing subsequence that converges to 0,knowing that inf{ x_{n} ,n ε N} =0??
prove if ## a_{2k} \rightarrow l ## and ## a_{2k-1} \rightarrow l ## then ## a_n \rightarrow l ## where ## a_{2k} ## and ## a_{2k-1} ## are subsequences of ## a_n ##
my attempt:
since: ## a_{2k} \rightarrow l ## then ## \forall \epsilon > 0 ## ##\exists N_1 \in \mathbb{R}## s.t. ##2k > N_1...
Homework Statement
a real sequence (x_{n}) is defined as follows: we take the elements in order (starting from
x0) to be
0, 1 , 0 , 1/10 , 2/10 ,... , 9/10, 1 0 , 1/100 ,2/100 ,..., 99/100 , 1 , 0 , 1/1000,...
So we take p for p = 0, 1, then p/10 for p = 0; ... 10, then p=100 for p =...
Homework Statement
2.11. Determine (explicitly) a convergent subsequence of the sequence in R2 given for n =
1; 2; : : : by
xn =(e^{n}sin(n\pi/7),((4n+3/3n+4)cos(n\pi/3))
I know that the Bolzano-weierstrass theorem says that every bounded sequence has a convergent subsequence. I...
SOLVED Prove that the functional sequence has no uniformly convergent subsequence -check
n \in \mathbb{R}, \ \ f_n \ : \ \mathbb{R} \rightarrow \mathbb{R}, \ \ f_n(x) =\cos nx
We want to prove that {f_n} has no uniformly convergent subsequence.
This is my attempt at proving that:
Suppose...
Hi all,
I was just looking at the U.S. electoral map, and I was wondering if there could possibly be a tie in presidential elections (the answer is probably no). I tried to think of an efficient algorithm to answer this question, but due to my limited intelligence and imagination, all I...
Question from my homework sheet. Can someone confirm I've got these correct.
Let (an)n∈N be any sequence of real numbers. Which of the following statements are true?
Give precise references to the results in the Lecture Notes for those which are true. Construct counter examples for those that...
Hi,
True or False: Every infinite sequence of natural numbers, who's terms are randomly ordered, must contain every possible subsequence of any length, including infinity.
For example, does the infinite and random sequence \small M of natural numbers require that the subsequence {59,1,6}...
Problem:
"Let x_1, ..., x_n be i.i.d random variables uniformly on [0,1]. Let X be the length of the longest increasing subsequence of x_1, ..., x_n. Show that E[X] \ge (1-o(1))(1-e^{-1}) \sqrt{n}."
Hi forum!
Using the Erdos' lemma I can only deduce that E[X] \ge \frac{1}{2} \sqrt{n}...
Homework Statement
Show that an increasing sequence is convergent if it has a convergent subsequence.
The Attempt at a Solution
Suppose xjn is a subsequence of xn and xjn→x.
Therefore \existsN such that jn>N implies |xjn-x|<\epsilon
It follows that n>jn>N implies |xn-x|<\epsilon...
Let $\{a_n\}$ be a sequence, and $\{a_{n_i}\}$ be any subsequence. Prove that if $\sum_{n=0}^\infty a_n$ is absolutely convergent, then $\sum_{i=0}^\infty a_{n_i}$ is absolutely convergent.
My attempt:
$\sum |\ a_n|$ is convergent.
$b_n=\left\{ \begin{array}{rcl}|a_{n_i}|\ &\text{for}& \...
Homework Statement
If \{a_{n}\}\in\mathbb{R} is Cauchy, \forall\epsilon>0,\exists a subsequence \{a_{k_{j}}\} so that |a_{k_{j}}-a_{k_{j+1}}|<\frac{\epsilon}{2^{j+1}}.
The Attempt at a Solution
Since \{a_{k_{j}}\} is Cauchy,\forall\epsilon>0,\exists N_{\epsilon} such that for j,j+1\geq...
I'm not sure if I am confusing myself or not, but a friend and I were trying to figure this out. Basically, I know that if a sequence is bounded, we are guaranteed at least one convergent subsequences. However, is it possible for a bounded sequence to have infinitely many of such subsequences?
Homework Statement
Let \{ f_{n} \}_{n=1}^{\infty} \subset C[0,1] be twice differentiable, and satisfying 0 = f_{n}(0) = f'_{n}(0) and \| f''_{n}\|_{\infty } . Prove that \{ f_{n} \}_{n=1}^{\infty} has a convergent subsequence.
Homework Equations
So since C[0,1] is a compact metric...
Not a homework problem.
I was reading this analysis book by Korner and in it there was a question about Bolzano-Weierstrass property in \mathbb{R} . it states
\text{Find a sequence} \ x_n \in [0,1], \text{such that given any x} \in [0,1], \text{we can find} \ n_1 < n_2 < ... s.t. \ x_n_j...
Homework Statement
Results
i) if (a_n) tends to L as n tends to infinity, then a_{n_r} tends to L as r tend to infinity
ii)if (a_n) tends to infinity as n tends to infinity, then a_{n_r} tends to infinity as r tend to infinity
using this result prove that
if (a_n) is an increasing...
Let X be the countably infinite product of closed unit intervals under the product topology. By Tychonoff's theorem, this space is compact. Consider the sequence \{ x_n \} , where x_k is the vector that is zero for all components except for the kth component, which is 1. Since this space...
Homework Statement
Let (sn) be a sequence in R that is bounded but diverges. Show that (sn) has (at least) two convergent subsequences, the limits of which are different.
Homework Equations
The Attempt at a Solution
I know that a convergent subsequence exists by...
Homework Statement
Asssume (an) is a bounded sequence with the property that every convergent subsequence of (an) converges to the same limit a. Show that (an) must converge to a.
Homework Equations
The Attempt at a Solution
If the subsequence converges to a we have , we have...
Hello all,
There is always a confusing question in my mind regarding sequence and subsequence, particularly in the field of probability theory and stochastic integration.
Given a sequence H^{n} which converges in probability to H, we know that there exists a subsequence H^{n_{k}} converging...
Homework Statement
Suppose x is an accumulation point of {an: n is a member of integers}. Show there is a subsequence of (an) that converges to x.
The Attempt at a Solution
I'm a little stuck on this one. I know that since x is an accumulation point then every neighborhood around x...
Homework Statement
Theorem: In a metric space X, if (xn) is a Cauchy sequence with a subsequence (xn_k) such that xn_k -> a, then xn->a.
Homework Equations
N/A
The Attempt at a Solution
1) According to this theorem, if we can show that ONE subsequence of xn converges to a, is that...
Homework Statement
Homework Equations
N/A
The Attempt at a Solution
I'm really not having much progress on this question. My thoughts are as shown above.
Homework Statement
Homework Equations
N/A
The Attempt at a Solution
Actually I read over this question for about 10 times already. But, I am not sure how to start. I know that I have to construct a subsequence one term at a time and show that it converges to L. But all the different...
Idea of a "subsequence"
I don't fully understand the idea of a "subsequence"...
1) Say we have an infinite sequence {an} = {1,2,3,4,...}
Then if we only take {1,2,3} with FINITE number of elements, is it a subsequence of an?
2) If an is an infinite sequence, is an a subsequence of an...
Homework Statement
Show that the lim sup of a bounded sequence is a limit of a subsequence.
Homework Equations
Sequence: Sn
Subsequence: Snk
The Attempt at a Solution
An existent lim sup means that at a large enough N, the subsequence could hug the bottom of the lim sup to within...
Homework Statement
Notation:
|a_n| is the absolute value of a_n.
(s_n) signifies a sequences; s_n signifies the value of the sequence at a particular n.
Problem:
Let n, k be arbitrary elements in N.
Let (a_n) be a sequence such that lim inf |a_n| = 0. Prove that there is a...
Homework Statement
Directly from the definition, for a sequence (s_n)_{n \in \mathbb{N}} \subseteq \mathbb{R} prove that if x > \limsup s_n , then x is not the limit of any subsequence of (s_n). (i.e. Do not use the fact that \limsup s_n is the supremum of the set of subsequential limits.)...
In Rudin's "Principles of Mathematical Analysis" he gives the strict definition for a subsequence as follows:
Given a subsequence {pn}, consider a sequence {nk}of positive integers, such that n1<n2<n3<... Then the sequence {pni} is called a subsequence of {pn}. If {pni} converges, its limit...
Homework Statement
Consider the sequence {x_k} = {(arctan(k^2+1),sink)} in R^2. Is there a convergent
subsequence? Justify your answer.
Homework Equations
Every bounded sequence in R^n has a convergent subsequence.
The Attempt at a Solution
To show {x_k} is bounded: The range...