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).
Homework Statement
Consider the sequence \left\{ x_{n} \right\}.
Then x_{n} is convergent and \lim x_{n}=a if and only if, for every non-trivial convergent subsequence, x_{n_{i}}, of x_{n}, \lim x_{n_{i}}=a.
Homework Equations
The definition of the limit of a series:
\lim {x_{n}} = a...
Homework Statement
Suppose that {Xn} is a sequence in R. Prove that Xn converges to a if and only if every subsequence of Xn converges to a.
Homework Equations
The Attempt at a Solution
Let e>0, choose N in N st n >=N implies |Xn-a| <e. Since a subsequence, nk, is in N and...
According to the Bolzano-Weierstrass theorem, a bounded sequence has a convergent subsequence. My problem is with the proof. Either I've got a bad textbook, or my reading comprehension is lacking. This is how it's formulated:
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Homework Statement
Prove:
Let (Xn) be a sequence in R (reals). Then (Xn) has a monotone subsequence.
Homework Equations
Def: Monotone: A sequence is monotone if it increases or decreases.
The Attempt at a Solution
I know it has something to do with peak points...that is there...
Homework Statement
Let {x_n} be a sequence in a metric space such that the distance between x_i and x_{i+1} is epsilon for some fixed epsilon > 0 and for all i. Can it be shown that this sequence has no convergent subsequence?
Homework Equations
None.
The Attempt at a Solution...
Hello, here is the exercise from the book:
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Let (a_{n})_{n=0}^{\infty} be a sequence which is not bounded. Show that there exists a subsequence (b_{n})_{n=0}^{\infty} of (a_{n})_{n=0}^{\infty} such that the limit \lim_{n\rightarrow \infty} 1/b_{n} = 0. (Hint: for each natural...
Hi,
Here is the question:
Prove that if the sequence {s} has no convergent subsequence then {|s|} diverges to infinity.
To me, this seems so easy, but I'm having a really hard time putting it down in a rigorous manner.
My thoughts are:
every convergent sequence has a convergent...
Does a subsequence only have to have "some" terms
This is an example from my text, which I do not understand.
Suppose (s_n)is a sequence of POSITIVE numbers such that inf{s_n | n in NaturalNumbers} = 0. The sequence need not converge or even be bounded , but it has a subsequence that...