Unterfolge

Unterfolge is a concept in mathematics referring to a subsequence. Given a sequence (a_n), a Unterfolge is another sequence (a_{n_k}) formed by selecting terms with strictly increasing indices n_1 < n_2 < … . In other words, a Unterfolge of (a_n) is obtained by deleting some terms while preserving the original order.

Notation and construction: If f: N → N is an increasing function, then the sequence (a_{f(k)}) is a Unterfolge of (a_n). Any choice of an increasing index sequence f yields a different Unterfolge.

Convergence and properties: If the original sequence (a_n) converges to a limit L, then every Unterfolge (a_{n_k}) also converges to L. Conversely, if a sequence has two Unterfolgen with different limits, the original sequence does not converge. A sequence may have infinitely many distinct Unterfolgen, of which some converge and others do not.

Examples: For a_n = 1/n, every Unterfolge converges to 0. For a_n = (-1)^n, the Unterfolge with even indices is constant 1 and the Unterfolge with odd indices is constant -1, illustrating how Unterfolgen can have different limits even when the original sequence may fail to converge.

Theorems and context: In metric spaces, the Bolzano–Weierstrass theorem states that every bounded sequence in R^n has a convergent Unterfolge. Unterfolge is closely related to the German term Teilfolge; in many texts the two words are used synonymously to denote a subsequence.

See also: subsequence, limit of a sequence, convergence criteria, Bolzano–Weierstrass theorem.