Limit of a Sequence

A sequence of elements $f_1,f_2,\dots$ in a vector space, or more generally in a metric space, is said to converge to the element $f$ if

$$\lim_{n\to\infty} f_n=f.$$

The element $f$ is called the limit of the sequence $\{ f_n\}$ . The meaning of this is that the distance

$$d(f_n,f)=\Vert f_n-f\Vert$$

between $f$ and $f_n$ can be made arbitrarily small by making $n$ sufficiently large. To summarize, a convergent sequence is one which converges to a limit.

Side Comment: It is easy to show that this limit is unique, a property which applies to all metric spaces.