You think the definition of "monotonic function" is more relevant to the meaning of "monotonically increasing sequence" than the definition of "monotone increasing" is?

[1,1,2], [1,1,1], and [1,2,2] are not monotone increasing. They're also not monotonic functions.

In all seriousness, the definition of monotonically increasing that I was taught is the same as exists in wikipedia:

"A function is called monotonically increasing (also increasing or non-decreasing), if for all x and y such that x <= y one has f(x) <= f(y).

https://en.wikipedia.org/wiki/Monotonic_function#/media/File...

This definition allows for 'flatness' in a graph, since the derivative does not change sign.

Or, from http://www-history.mcs.st-and.ac.uk/~john/analysis/Lectures/... :

    A sequence f(n) is monotonic increasing if f(n+1) ≥ f(n) for all n ∈ N.
    The sequence is *strictly* monotonic increasing if we have > in the definition instead of ≥.

You can draw that contrast (increasing vs strictly increasing), but what I was taught was to contrast increasing functions/sequences with nondecreasing functions/sequences.

A book or paper will make it clear what they mean by "increasing" by using the definition. There, it doesn't matter at all -- they could just as easily coin new words, since they immediately give the full definition. But the people hanging around this thread, telling people who are using a very common definition of "monotonically increasing" that (in paraphrase) "I hate to be pedantic, but you've made a mistake, in that I would have phrased that differently" have failed to contribute anything or to be pedantically correct. There's no case to be made that, if I say a "monotonically increasing sequence" must be increasing rather than nondecreasing, I've made a terminological mistake. This is a term with different definitions in different treatments.

Thanks for your perspective. It's interesting that there appears to be ambiguity in the terms and a diversity in what is being taught.