Can a sequence be neither decreasing nor increasing? Given the definition of a increasing sequence:
"A sequence $(a_n)_{n\in\mathbb{N}}$ is increasing if for all $n\in \mathbb{N}$,    $a_n\le a_{n+1}$."
My question is: by this definition isn't the sequence $(1,1,1,1,1,\dotsc)$ an increasing sequence then?
 A: If your definition reads

Definition: A sequence $(a_n)$ is increasing if $a_n \le a_{n+1}$ for all $n$,

(or something similar), then the sequence $(1,1,1,\dotsc)$ is an increasing sequence.  It is increasing precisely because it satisfies the property which defines an increasing sequence (per the definition given).
This may not feel right, as our intuition from natural language is that this sequence is constant, and therefore not increasing.  There are two bits of advice that I would give (the first when you have to deal with other people's exposition, the second when you are doing your own work):

*

*Just get used to it.  There are many terms in mathematics which come from natural language, but which don't mean the same thing in mathematics as they do in vernacular English.  The terms "open" and "closed" are a big bugaboo for intro to topology students, for example.  As a mathematician, you should learn to get used to words that are given technical definitions which contradict plain English interpretation (and, indeed, may contradict each other, as different authors may define the same term differently.


*Find another term.  In his series on analysis, Simon comments early on that weak vs strict inequalities are possibly ambiguous.  He therefore declares that, in his text, "increasing" always means "nondecreasing" (and, if I recall correctly, he also declares that "positive" means "nonnegative"; the point is that he doesn't want to have to say things like "nonnegative nondecreasing").  A similar strategy may help you:  if you don't want to call a constant sequence "increasing", use the words "nondecreasing" (for a sequence where $a_n \le a_{n+1}$), and use "increasing" (or even "strictly increasing") for sequences which satisfy $a_n < a_{n+1}$.
Simon, Barry, Real analysis. A comprehensive course in analysis, part 1, Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-1099-5/hbk). xx, 789 p. (2015). ZBL1332.00003.
