Monotonicity of a sequence
I need to determine whether the sequence , where , is decreasing or increasing. To do that, I can first suppose that it's e.g. increasing, and then see if the relation holds for every . In other words, if the inequality is true, then is also , and if it isn't, then and the sequence is decreasing. This is where I got stuck, for no matter how I tried to solve this inequality (squaring both sides; multiplying whole inequality by both denominators etc), I can't seem to arrive at the solution.
To give you a better idea of the way I am supposed to solve this, here is an easier example:
determine whether the sequence , where , is decreasing or increasing.
We can start by assuming that is increasing; in that case is true if and only if , which is true, and therefore the starting assumption ( ) is also true, and is an increasing sequence.
So, basically, I should somehow arrive from this expression: to a statement from which the verity of the starting assumption (i.e. ) could be inferred.