Could give me an informal, but detailed explanation of what Cauchy sequences are? So far, just reading at the book, I have got the following lemmas: (1) every convergent sequence is a Cauchy sequence (2) A real sequence is convergent if and only if it is a Cauchy sequence (Cauchy convergence criterion). (3) In wikipedia, I read that: "a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses [...] the criterion for convergence depends only on the terms of the sequence itself, as opposed to the definition of convergence, which uses the limit value as well as the terms".
Thus, if I understand what I report here, a Cauchy sequence is just another way to define any convergent sequence. However, since that I am at the beginning of my studies and I have problems in dealing with formal, mathematical language, I would really appreciate if you can give me a more informal explanation of Cauchy criterion and example(s) of Cauchy sequence.
Also I would really appreciate if you can introduce me, more gently than books do, to the formal definition of both convergent sequence defined by the limit and Cauchy sequence.
With the information you give me about Cauchy theorem (I think "Cauchy criterion" and "Cauchy theorem" are 2 ways to say the same thing) I would like to prove the convergences of:
$$a_n = 1 / 2^n$$ 
and 
$$a_n = \sin (n) / n$$
PS. If you want to show me also the proofs, with an explanation of that, I would be also grateful because I have an imminent deadline. Oterwhise, it is fine if you can provide answer to the previous question(s). Thank you!
PS. could someone provide me proof at least for the second sequence, please? I have the impression that the second one is slightly more difficult to prove
 A: The confusion you're dealing with is because the real numbers have a property called "Completeness".   In general,   convergent implies cauchy in any space.  However,  if your space is complete,  it becomes an if and only if, so the two terms become equivalent.
To give you an idea of a space that is NOT complete and how the Cauchy criterea is different from being convergent,  look at the rational numbers.    Define $a_n$ to be the first $n$ digits of the decimal expansion of $\sqrt 2$
Now,  this sequence is Cauchy, because for any error tolerance, i.e.,  $\epsilon >0$ we can find a $N$ such that for all $n\ge N,m\ge N$,  $|a_n-a_m|<\epsilon$  (Just take N to be the number of 0s in the decimal expansion of $\epsilon$ until you hit a nonzero number).
However, it does not converge!  To see this, realize that the number it would converge to is $\sqrt 2$,  but $\sqrt 2$ is not in the rational numbers. So,  there is no number $L$ IN THE RATIONALS such that $\lim a_n=L$.
As a side note,  every incomplete metric space can be embedded into a unique (up to isomorphism) complete metric space, formed by basically "filling in the holes" like this.   There's a method to it,  but the machinery is not very important right now.
EDIT:  A short explanation of what Cauchy sequences are are they are sequences where the terms eventually get arbitrarilly close TO EACH OTHER.   A convergent sequence is one where the terms get arbitrarilly close to a specific number.   In a complete metric space, there are no holes,  so if the terms all get close to each other, there must be a specific number they get close to.   In an incomplete metric space like the rationals,  you can have cauchy sequences that approach one of these holes (Which in this case, are the irrationals)
A: I am going to only answer your last query in your post script, and believe that you can get adequate explanations on Cauchy sequences from Bartle and Sherbert's Introduction to Real Analysis, or W. Rudin's Principles of Mathematical Analysis. Now, consider sin(n)/n = y(n). Since -1 <= sin(n) <= +1, for all natural n, -1/n <= y(n) <= 1/n, so by the Sandwich Lemma, since {-1/n}, {1/n} converge to 0, y(n) converges to 0. Since y(n) converges, and y(n) is a real sequence, it is Cauchy.
P.S: Please do not assume that Cauchy sequences are always convergent in all spaces. This is not true.
