I presume you asked that question because you expected me to start with the usual prove a_k - a_{k-1} >= 0 for all k>1, to which you would then argue that I started with the required result.
However. the example you have here is very different to the above example you gave above with the iff arguments going backwards.
To me, the equivalent of starting with the given identity then playing around with it until you reach a trivial result, would be something like starting your proof with:
k} \geq \frac{2^{k+1}}{(3^{k+1}+1)(k+1)})
And then working your way down.
I would rather see something like:
"Consider a_k - a_{k-1}" etc etc, THEN working with it. I know it seems very similar to the above, but the example you provided doesn't really leave much room to differentiate between the two methods.
You can use the given identity to have an idea of where to START the proof, but not to always actually use the given identity, then work your way down to a trivial result.
So suppose wanted to prove x^2 + y^2 >= 2xy.
Sure, you could do what you had there with the iff going backwards etc, but I wouldn't think it's as nice as say starting from (x-y)^2 >= 0, then acquiring it because (I know this is a basic example but ofc there are harder ones) thinking of starting with (x-y)^2 >= 0 requires a certain level of ingenuity, whereas anybody can just re-arrange the given expression and play around with it until a trivial result is held, then claim end of proof.
Both are correct of course, and indeed for many inequality problems it's easier to start with the given result, then work down to trivial result (usually something like (x-y)^2 >= 0), but I don't think using that method really develops that 'sixth sense' in Maths that I think is so important.
When I say '6th sense', I mean that feeling when you just 'know' what to do but can't really explain it too easily, I am sure you know what I mean.
I see the learning experience beyond just 'doing the inequality questions to get marks' because that kinda defeats the purpose of the education of Mathematics. I see it more as 'developing your intuition', which I think would be MUCH better done by starting with a known result (or using the required identity to guess where to start), then deducing the answer, as opposed to starting with the answer then deducing something obvious.
I thought my way was right? Unless that's only for integrating, in which case it would stilll be right wouldn't it since you can just go backwards through my proof by starting at
for x>0 and integrating.
