Monotonic sequence definition Monotonic sequence definition
i) A sequence is monotonic if successive terms are nondecreasing, or if they are nonincreasing.
But in another book, I see that:
ii) A sequence is monotonic if it is decreasing, or increasing .
Now, if I take the def(i), I will get that:
$a_n=1,-2,3,-4,...$
Is nondecreasing, and nonincreasing, so $a_n$ is monotonic,
And if I take the def(ii), I will get that, $a_n$ is not monotonic because it’s not increasing and not decreasing.
What’s the wrong?
 A: You're interpreting the first definition wrong. When they say "if successive terms are nondecreasing, or if they are nonincreasing", then they mean that the same word (either nondecreasing or nonincreasing) should apply to all the terms. You aren't allowed to switch which word you apply (if that were the case, they would've said something like "if successive terms are either nondecreasing or nonincreasing", changing whether the "or" is inside the "if each term" clause, or between two "if each term" clauses). In your example sequence, you have to switch.
That being said, nonincreasing and nondecreasing is some of the most confusing and ambiguous terminology I've seen in mathematics, and I wish no one used it.
"Nondecreasing sequence" means a sequence which never decreases, not a sequence which is not a decreasing sequence. However, that's what "increasing sequence" means in modern mathematics lingo. A sequence which actually increases from term to term is a strictly increasing sequence.
