Hey guys,
Just wondering would you be able to help me with this question. I know of a proof using calculus, but I think there may be another method.
for
(Edited) Hmm, I thought I had come up with a nifty proof, but seems now that I wasn't able to avoid calculus.
This is of the form
where
Now all we need to establish is that
In other words, we need to show that is increasing.
Now, I had originally written:
"Clearly, the numerator of grows faster than the denominator, thus is increasing and the proof is concluded."
But I don't think that's quite good enough, and now I'm not sure how to prove rigorously that is increasing without derivatives... maybe someone else can see a way?
EDIT: I'm not confident about the following method, but perhaps it is valid:
Let .
Since preserves order (is monotonic), applying to both sides of an equation will also preserve order. So:
Let . Then it is sufficient to show that is increasing, in order to prove that is increasing.
Now let . This is again order preserving, and we have:
It is clear that is increasing, therefore is increasing and the proof is concluded.
I'm bothered by the result because, when I took derivatives, I found that when , and nowhere in the above steps did I utilize the restriction .
Ah well, I've edited this post about a million times, I'll leave it in the hands of other, smarter people.