Partial sequence

In mathematics , a subsequence of a sequence is a new sequence that arises when members of the sequence are omitted from the original sequence. A finite number of terms (especially none at all) or an infinite number can be omitted. Unless a finite partial sequence is explicitly mentioned, an infinite partial sequence is usually meant again in the case of an infinite partial sequence.

A partial sequence can be formed from the sequence by only considering the elements , with a strictly monotonically growing infinite sequence. ${\displaystyle (a_{n})}$ ${\displaystyle (a_{n_{k}})_{k\in \mathbb {N} }}$ ${\displaystyle n_{1}<n_{2}<n_{3}<\dotsb }$

${\displaystyle (a_{n})}$is itself also a subsequence of . ${\displaystyle (a_{n})}$

Examples

• Consequence . Partial sequence with : ${\displaystyle a_{n}=(-1)^{n}}$ ${\displaystyle n_{k}=2k}$ ${\displaystyle (a_{n_{k}})=(1,1,1,\dotsc )}$
• Consequence . Partial sequence with : ${\displaystyle a_{n}=n}$ ${\displaystyle n_{k}=k^{2}}$ ${\displaystyle (a_{n_{k}})=(1,4,9,16,\dotsc )}$

Sequence compact space

According to the Bolzano-Weierstrass theorem , every bounded infinite real number sequence has at least one convergent subsequence. In general, a topological space is called sequence- compact if it has the property that every sequence has at least one convergent subsequence.

convergence

If a sequence is convergent to , then each subsequence also converges to the same limit value . Conversely, even if any sub-sequence to the same limit converges, that the sequence against converges. ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$ ${\displaystyle a}$ ${\displaystyle (a_{n_{k}})_{k\in \mathbb {N} }}$ ${\displaystyle a}$ ${\displaystyle (a_{n_{k}})_{k\in \mathbb {N} }}$ ${\displaystyle a}$ ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$ ${\displaystyle a}$

In any topological space, the set is even considered that an impact if and against if every subsequence converges a part subsequence includes opposing converges. The meaning of this theorem is, firstly, that it is helpful for many proofs of convergence in consequent compact spaces. Second, this sentence provides a criterion as to whether a concept of convergence can be described by a topology; the point-wise convergence almost everywhere of a sequence of functions does not, for example, satisfy this theorem and therefore cannot be described by a topology. ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$ ${\displaystyle a}$ ${\displaystyle (a_{n_{k}})_{k\in \mathbb {N} }}$ ${\displaystyle (a_{n_{k_{l}}})_{l\in \mathbb {N} }}$ ${\displaystyle a}$

Web links

Wikibooks: Math for Non-Freaks: Partial Sequence  - Learning and Teaching Materials

literature

• Konrad Königsberger : Analysis 1 . Springer, Berlin 2004, ISBN 3-540-41282-4