Sequential Limit of a Sequence [Xn] Formed on Closed Subset of R: Understanding the concept of limits in sequences

Abstract

Thelimit of a sequence is the valuethat the terms of a sequence "tend to". If such a limit exists, thesequence is called convergent. Asequence which does not converge is said to be divergent. The limit of a sequence is said to be the fundamentalnotion on which the whole of analysis ultimately rests. Limitscan be defined in any metric or topologicalspace, but are usually first encountered in the realnumbers. the limit of the sequence if the following conditionholds: Foreach realnumber , there exists a naturalnumber such that, for every natural number , we have . Inother words, for every measure of closeness , the sequence's terms areeventually that close to the limit. The sequence is said to converge to or tend to the limit , written or . Ifa sequence converges to some limit, then it is convergent; otherwise it is divergent.