Given a sequence $a_n$ defined in a recursive manner, the methods I know to prove if the sequence is increasing are:
1) observe if $a_{n+1} - a_n > 0 \ \forall n.$
2) take $\frac{a_{n+1}}{a_n}$ and observe if it is $> 1$
3)procede by Induction.
4)take the first derivative (if possible) and see if it is always positive. Also check the starting element of the sequence.
My questions are: If unsure about witch method to use first is there a hierarchy of usefullness (as happens with the root and ratio test for series) ?
Are there other methods to prove a sequence is increasing?
Is anyone of the one ones I mentioned redundant (if I tried one method it would be useless to try with an other specific method)?