This is, imo, an odd-worded question: is D, the domain, unbounded above, but then f is "eventually" (what does this mean mathematicalwise?) monotone and bounded? Well, if it has some more or less intuitive meaning, then f is bounded so how can it go to oo??
Try to word out the problem more clearly, perhaps that'll help to undestand what's going in here.
If f is defined on a set then f is eventually bounded on D if such that ,
If f is defined on a set then f is eventually monotone on D if and , either or
I believe if one looks at the function as a sequence the word 'eventually' is definitely defined in mathematics, as in 'eventually' monotone sequences where
for some but where for some smaller n and m.
Well, those are the definitions of bounded and monotone. The questions everyone's been asking are:
1) What is this "eventually" business?
2) How can anything bounded ever have a limit approaching infinity?
Until some things are cleared up, no one is going to be able to help.
It seems that a sequence a_n can be monotone for n>N but for n<N it may not be. Hence this sequence would be eventually monotone rather than strictly monotone.
like this sequence: