TITLE: What is the difference between Cauchy and convergent sequence?
QUESTION [30 upvotes]: I am really confused. I will appreciate if somebody can help me to define the difference between Cauchy and convergent sequence. 
Many thanks.

REPLY [27 votes]: When you have a convergent sequence: $x_n\to\xi$, then for large $n$ all points $x_n$ are near $\xi$, and by the triangle inequality they are then near each other as well – in formal language: The $x_n$ automatically form a Cauchy sequence; no big deal.
But:
In some metric spaces $X$, among them ${\mathbb R}$, ${\mathbb R}^n$, and ${\mathbb C}$, the converse also holds: A Cauchy sequence is automatically convergent. (Such spaces are called complete.) This property of spaces is all-important for the following reason: In order to test a sequence $(x_n)_{\geq1}$ for Cauchy-ness you only have to look at the given $x_n$ themselves, and you don't have to know the limit beforehand. When the sequence passes this "finitary" test you know for sure that it has a limit $\xi\in X$. 
In all, the so-called Cauchy criterion (for real sequences, say) is not a little proposition about convergence, but a deep theorem about the fine structure of ${\mathbb R}$.