Talk:Exponent Combination Laws/Product of Powers/Proof 2/Lemma

Better result
The original, more general inequality is probably true. However, the proof is not coming to me right now. I think the product of powers proof can proceed with this weaker version. --Keith.U (talk) 01:09, 3 July 2016 (UTC)


 * I'm still having trouble with this lemma. The setup looks overcomplicated, and further abstraction is needed.


 * The gist of it is that (suitably restricted):
 * $\left\vert{x_1 - x_2}\right\vert < \epsilon \land \left\vert{y_1 - y_2}\right\vert < \epsilon \implies \left\vert{x_1 y_1 - x_2 y_2}\right\vert < \epsilon \left({2 M + 1}\right)$
 * where $M = \max \{x_1, x_2, y_1, y_2\}$.


 * Then all you need to do is invoke the monotonicity of the exponential at the end.


 * Other problems:
 * it is mentioned that $a^r$ and $b^r$ are restricted to $I$, but $b^r$ never appears.
 * no mention of what $x$, $y$ and $s$ are.


 * Is this proof being crafted out of whole cloth? If so, you might want to pass it by an authority before posting it up here. If not, then how is this approached in your source work? --prime mover (talk) 08:45, 3 July 2016 (UTC)


 * Good point with the abstraction.


 * I was hoping I could get away with not mentioning the nature of the variables, with the tacit understanding as to whether they're real or rational. I guess I'm still getting used to the house standards.


 * I looked hard for a rigorous development of the exponential from this regard, but all the sources I found just give (very) general outlines of proofs. So yes, this proof is my own. In retrospect, I should have added the proofread tag. --Keith.U (talk) 09:37, 3 July 2016 (UTC)