Subsequence

In mathematics, a subsequence of a sequence a new sequence is produced when the follower members are omitted from the original sequence. It can be omitted or infinitely many finite number of terms ( no particular ). Except where specifically speak of a finite subsequence is again usually meant an infinite subsequence in an infinite sequence.

A subsequence can be formed from the sequence by using only the elements be considered, with a strictly increasing infinite sequence is.

Is itself a subsequence of.

Examples

• Consequence. Subsequence with:
• Consequence. Subsequence with:

Compact sequence space

By the theorem of Bolzano- Weierstrass every bounded infinite sequence of real numbers has at least one convergent subsequence. General is called a topological space sequentially compact if it has the property that every sequence has at least one convergent subsequence.

Convergence

If a sequence is convergent against so every subsequence converges to the same limit. Conversely, even if every subsequence converges to the same limit that the sequence converges to.

In any topological space even have the theorem that a sequence converges if and only against, if every subsequence contains a partial subsequence that converges to. The meaning of this sentence is, first, that it is helpful in many convergence proofs in sequentially compact spaces. Second, this theorem gives a criterion of whether a concept of convergence can be described by a topology; the pointwise convergence almost everywhere of a sequence of functions not met, for example, this sentence and therefore can not be described by a topology.