Limit

A sequence can be defined as a function on the domain of natural number like $$1, 1/2, 1/3 ... 1/n$$. This sequence approach to the 0, but never touch the 0. However, people can not take their desire to link the sequence and the 0. Because $$\infty$$ is not a member of the natural number even real number, another concept is necessary to link the sequence and the 0. It is the limit. $\lim{n\to\infty}$

The above sequence approach to the 0. But does all sequences approach to some points? What if the sequence is $$1/n$$ if $$n$$ is not multiple of 100, 0.001 if n is multiple of 100.

Its approach to zero except at every multiple of 10. The $$\epsilon$$ is used for the definition of limit to exclude this example.

The sequence $$S$$ is converges the limit $$a$$ if for every positive $$\epsilon$$, natural number $$N$$ is present such that $$\vert a-Sn \vert <\epsilon$$ is true in every $$n>N$$. Otherwise, the limit is not defined and the sequence is divergent.

In topological space, the $$\epsilon$$ becomes the neighborhood.

Jun Kang
Clinical Assistant Professor of Hospital Pathology

My research interests include pathology, oncology and statistics.