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.