The correct approach for any data modeling problem is to think in terms of entropy. Each subsequent approach should minimize entropy, until you reach diminishing returns.
The sequence is not monotonically increasing. It's strictly increasing. If you test [1, 1, 2] or [1, 1, 1] or [1, 2, 2], you'll get "No" answers even though those sequences are monotonically increasing.
[1,1,2], [1,1,1], and [1,2,2] are not monotone increasing. They're also not monotonic functions.
"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 ≥.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.