Being $\{a_n\}_{n\in N }$ a succession of real numbers which is bounded and NOT monotonic, can the succesion converge?
To answer the question I must find one succesion $a_n$ that converges, For example:
$$a_n = (-1)^n\left(\frac{1}{n}\right)$$
Which converges, is bounded and is not monotonic (as far as I know) then the answer to the question is YES?
Yes. Since the question is whether a certain thing can happen and since you found a situation in which that thing does happen, the answer is affirmative.