Cauchy sequence and convergent sequence There are two theorems in my book saying that:
"Convergent sequences are Cauchy sequences" and "Cauchy sequences are convergent sequences." So I am wondering what's the difference between Cauchy sequence and convergent sequence. Thanks!
 A: First of all, they are conceptually different ideas: it turns out that the same sequences satisfy those concepts.  (BUT see below.)
Analogously, you can say that an equilateral triangle is one with three equal sides, and an equiangular triangle is one with three equal angles.  These concepts are expressing different ideas, but it turns out that they describe the same triangles.
The second and more important point is that your book is probably, implicitly or explicitly, discussing sequences of real numbers.  (Or perhaps complex numbers.)  Suppose, on the other hand, that the only numbers we know about are rationals.  Then it is still true that all convergent sequences are Cauchy; however it is not true that all Cauchy sequences are convergent.  For example,
$$\{3,\,3.1,\,3.14,\,3.141,\,3.1415,\,3.14159,\ldots\}$$
is a Cauchy sequence in $\Bbb Q$ which does not converge (does not have a limit) in $\Bbb Q$.  Note that you can't say the limit is $\pi$ because we are assuming the only numbers we know about are rationals, and $\pi$ is not rational.
To continue the triangle analogy: "equilateral" and "equiangular" do not describe the same shapes if we are talking about, say, quadrilaterals.  (Or quadrangles!)
A: The difference is in their definitions.
A convergent sequence
$(a_i)_{i=1}^{\infty}$
is one for which there
exists a limit $L$
such that,
for any $\epsilon > 0$
there is a $N(\epsilon)$
such that
if $n > N(\epsilon)$
then
$|a_n - L| < \epsilon$.
A Cauchy sequence
is one such that
for any $\epsilon > 0$
there is a $N(\epsilon)$
such that
if $n, m > N(\epsilon)$
then
$|a_n - a_m| < \epsilon$.
The fact that evey Cauchy sequence
is convergent is non-trivial to prove.
The advantage of a sequence
being Cauchy is that
you do not have to specify a limit.
