Posloupnost

Posloupnost, in Czech mathematical usage, denotes a sequence: an ordered list (a_n) of elements from a set X indexed by natural numbers. Most commonly X is the real numbers, but the concept applies to any set. The n-th term is a_n, where n is a positive integer (sometimes starting at 0). A finite sequence contains finitely many terms, while an infinite sequence continues indefinitely.

Two familiar examples are arithmetic progressions, where a_n = a_1 + (n−1)d, and geometric progressions, where a_n = a_1 r^{n−1}. Beyond these, a sequence may exhibit monotonicity (nondecreasing or nonincreasing), boundedness (bounded above and/or below), or be used to form subsequences by selecting terms with increasing indices.

Limit and convergence: A real sequence (a_n) converges to L if, for every ε > 0, there exists N such that |a_n − L| < ε for all n ≥ N. Convergent sequences are necessarily bounded. A sequence is called Cauchy if, for every ε > 0, there exists N such that |a_n − a_m| < ε for all n, m ≥ N.

Role in mathematics: Sequences are foundational to the study of limits, series, continuity, and function approximation. They are used to model processes over time in discrete settings and serve as a simple bridge between discrete and continuous analysis.