# 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}

${\ displaystyle (a_ {n})}$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}$