What is the point of the term "monotonic" when analyzing sequences? If my understanding is good, a sequence is monotonic if it increase or decrease.
In my homework, I was asked to indicate if some sequences were increasing or decreasing, and also to specify if it was monotonic or not.
To me, it seems useless to specify if it is monotonic or not as specifying whether the sequence increase/decrease or not will automatically indicate it.
Now my question is, is there something that I misunderstood about monotonic sequences, or specifying it is just useless/was just in the goal of seeing if we properly understand the concept ?
 A: Typically,  monotonic means either increasing or decreasing, as you say.    The terminology for monotonically increasing is 'isotonic',  and for monotically decreasing is 'antitonic'
A: Definition: A sequence $\{a_n\}$ is said to be


*

*increasing if and only if for all $n,m\in\mathbb{N}$, with $n<m$, we have $a_n\leq a_,$.

*eventually increasing if and only if there exists $n^*\in\mathbb{N}$ such that for all $n,m\in\mathbb{N}$, with $n^*\leq n<m$, we have $a_n\leq a_m$.

*strictly increasing if and only if for all $n,m\in\mathbb{N}$, with $n<m$, we have $a_n<a_m$.

*eventually strictly increasing if and only there exists $n^*\in\mathbb{N}$ such that for all $n,m\in\mathbb{N}$, with $n^*\leq n<m$, we have $a_n<a_m$. 


In a similar manner, the terms decreasing, eventually decreasing, strictly decreasing, eventually strictly decreasing may be defined.
If a sequence is either increasing or decreasing, then we call it monotone. Similarly, a strictly monotone sequence is strictly increasing or strictly decreasing. We can also define eventually monotone and eventually strictly monotone sequences in a similar manner. 
If you want more than that though and your question is more "philosophical," then I'm afraid there's not much there. Consider what it means for a function to be one-to-one, onto, and one-to-one and onto, that is, what it means for a function to be injective, surjective, and bijective. 
A: Every monotonic limited sequence converges. This is an application of the term monotonic.
