Why a particular type of definition for increasing sequence is chosen? I am asking the following question in the context of first-level analysis course.
A increasing sequence is generally defined as: A sequence is increasing if and only if $a_{n+1} \geq a_n $ for all n.
Would it be wrong if I define increasing sequence in the folowing manner: A sequence is increasing if $m \geq n $ then $a_m \geq a_n$ and vice-versa.
I really want to know if there exists a conceptual difference between the above two definitions or are they conceptually equivalent.
Thank you for your responses. 
 A: The two statements:

A sequence is increasing if and only if $a_{n+1} \geq a_n \quad \forall n$.

and

A sequence is increasing if and only if $a_m \geq a_n$ whenever $m \geq n$.

are equivalent.
Given the first definition, for any $m \geq n$ we can say $a_m \geq a_{m-1} \geq \dots \geq a_{n+1} \geq a_n$ so $a_m \geq a_n$ by transitivity.
Given the second definition, we simply take $m = n+1$.
Though both definitions are equivalent, I would suppose that we take the first definition because it is more direct to the salient property of an increasing sequence. Namely, that each term is greater than or equal to the last.
A: There is no difference, Your definition is more general because if you take $\color{red}{m = n+1}$  you will basically reach the same definition that you read in your book. That is $$\color{blue}{\large{a_{n+1} \geq a_n \space \forall \space n}}$$
You kinda did an induction of the original definition because from $\color{brown}{a_{n+1} \geq a_n \space \forall \space n}$ you also get that $\color{green}{a_{n+2} \geq a_{n+1} \geq a_n \forall n}$ and from that you get that $\color{magenta}{a_{n+3} \geq a_{n+2} \geq a_{n+1} \geq a_{n} \forall n}$
and eventually you reach your definition
A: I assume the "vice versa" is an error.
Although the candidate definitions
$$ \forall n : a_n\le a_{n+1} \tag{1} $$
and
$$ \forall m,n : m\le n\implies a_m\le a_n \tag{2} $$
are logically equivalent in your context, there is definitely a conceptual difference.    One way to expose it is to try using these definitions in more general settings.  So, suppose we wanted to consider a kind of "pseudo-sequence" where the indices aren't natural numbers, but some other kind of object (maybe real numbers, or pairs of integers, or lines in space, or other sequences, or...), and we want to define the term "increasing" for these pseudo-sequences.  If we try to adapt definition (1) to this new setting, we'll need some analogue of "$n+1$" for the indices; one way to think of "$n+1$" is that it represents the "next" natural number after $n$, so we'll want there to be a clear notion of "next" for our indices.  This probably calls for a set of indices which is well-ordered.  (We'll think of "$n+1$" as the minimum element of the set $\{x : x>n\}$.)  If we use definition (2), however, we only need an analogue of $\le$, which formally could be any binary relation — though if we want our pseudo-sequences to behave a bit more the way sequences do in analysis, we'll probably want the indices to form at least a partial order, and maybe a directed set, which are still much weaker conditions than well-ordering.
More practically, in problem-solving you'll often find it's easier to prove increasingness in form (1), but to use it in form (2).
