Cauchy Sequence

It is clear that the elements $f_1,f_2,\dots ,f_m,\dots ,f_n,\dots$ become closer and closer in some sense. In fact, from the triangle inequality for a vector space

$$\Vert f+g\Vert\le\Vert f\Vert +\Vert g\Vert$$

one finds that

$$\Vert f_n-f_m\Vert = \Vert (f_n-f)+(f-f_m)\Vert$$ (13)

$$\le \Vert f_n-f\Vert +\Vert f-f_m\Vert\to 0~~\textrm{as}~m,n\to\infty$$ (14)

or more generally

$$d(f_n,f_m)\le d(f_n,f)+d(f,f_m)\to 0 ~~\textrm{as}~m,n\to\infty$$

in a metric space.

Consequently

$$\lim_{n,m\to\infty} \Vert f_n-f_m\Vert =0~~\textrm{in~a~vector~space}$$

or

$$\lim_{n,m\to\infty}d(f_n,f_m)=0~~\textrm{in~a~metric~space}\,.$$

A sequence $\{ f_n\}$ whose elements satisfy this limit condition, i.e., whose elements get arbitrarily close together for sufficiently large $n$ and $m$ , is a Cauchy sequence. Thus

$$\{ f_n\}~~\textrm{has~a~limit}~~\Rightarrow\{ f_n\}~~\textrm{is~a~Cauchy~ sequence}\,.$$

Thus every convergent sequence is a Cauchy sequence, i.e., ``every convergent sequence also converges in the Cauchy sense''.