Although it is very easy, you also need to prove that for all .
It is a little 'surpising' the fact that You have a total of 973 posts and none of them is written in Latex!... the question is not minor because what You have written can cause misundestanding!...
If the 'recursive relation' is...
, (1)
... then is and the sequence is neither increasing nor decreasing...
If the 'recursive relation' is...
, (2)
... then is...
(3)
... and the (3) is negative so that the sequence is decreasing and it finishes at the n for which is ...
There are some other 'possible interpretations' but in any case the 'advatages' of the use of Latex are fully evident!...
Kind regards
I would have interpreted that as !
To prove, by induction that this is a strictly decreasing sequence, that is, that for all n, note that and that so the statement is true for n=1.
Assume that for some k. Then but since , . That is, and we are done.
If it is, in fact, , then the sequence is neither increasing nor decreasing- for all n!