convergent sequence definition and example

One can think of it as a function N → R, where N is the set of positive integers. Our focus here is to provide a rigourous foundation for the statement “sequence ( a n) → L as n → ∞”.ĭefinition (Convergence). Definition of ConvergenceĪ sequence in R is given by ( a 1, a 2, a 3, …), where each a i is in R. Thus, L = sup(S) iff (i) L is an upper bound, (ii) for each ε>0, there exists x in S, x > L-ε. Since L-ε is not an upper bound, there exists x in S, x > L-ε. the set of real (or rational) numbers 0 0). A set may be upper-bounded without possessing a maximum, e.g.If min( S) exists, then so does inf( S), and min( S) = inf( S).

If max( S) exists, then so does sup( S), and max( S) = sup( S).

The maximum of T is called the infimum of S, denoted inf(S).

Let S be lower-bounded, and T the set of lower bounds of S.

The minimum of T is called the supremum of S, denoted sup(S).

Let S be upper-bounded, and T the set of upper bounds of S.

If S is both upper- and lower-bounded, we just say it is bounded.

If is a lower bound of S, we call it the minimum of S, denoted min(S).

If S has a lower bound, we say it is lower bounded.

A lower bound of S is an element such that x ≤ y for all.

If is an upper bound of S, we call it the maximum of S, denoted max(S).

If S has an upper bound, we say it is upper bounded.

An upper bound of S is an element such that x ≥ y for all.

Suppose X = R or Q and S is a non-empty subset of X. We shall now describe this in a more rigourous manner.įirst some basic definitions. One should have a mental picture of the set of rational numbers Q having “gaps” in its order structure, while R “fills up” these gaps.

continuity, differentiation, sequence convergence etc. It provides a rigourous foundation of concepts which we usually take for granted, e.g. Much of analysis deals with the study of R, the set of real numbers.

Convergent sequence definition and example