The answer is No. In other words, such a sequence

does not necessarily eventually decrease.

To start with, consider the sequence

defined by

for

Then

and you can check by multiplying out the fractions that

for all k. Thus there are infinitely many values of n for which

That example does not quite answer the original question, because the sequence

is not strictly increasing (being constant throughout the interval

). However, in principle there is no difficulty in adjusting the sequence so as to make it strictly increasing. For each k, you can make

very slightly smaller, without disturbing the property that

. Then you can interpolate the values of

linearly for

so as to ensure that the sequence increases strictly.