Significance of the sequence (1/n)
Significance of the sequence (1/n) M. Velrajan We know that the sequence 1/n → 0 and it follows from Archimedean Property of Real Numbers : If x, y are real numbers with x > 0 then there is a positive integer n such that nx > y. In fact, it follows as the particular case x = ϵ > 0 and y = 1 of the Archimedean Property. The sequence 1/n → 0 and some of the results that follow from it are to derive many results not only in real sequences and series but also in metric spaces and topological spaces. We highlight some of the significance of 1/n → 0. First we recall some of the important results in real sequences that follow from 1/n → 0 and then their uses in metric spaces and topological spaces. 1. For all real k, the sequence k/n → 0 and hence, for any real number x, the sequence (x + k/n) → x . ( since | (x + k/n) - x | = | k/n | ) In general, if a n → a then for all real k, the sequence (a n + k/n) ...