I refer to your explanation of Cauchy's Bound in the Polynomials chapter of Precalculus 3.
With some theorems, such as the Rational Zeros Theorem, I had already worked out a rationalisation of the theorem that satisfied me of its truth. In the case of Descartes' Rule of Signs, I couldn't find an explanation that was "accessible" to me, but I eventually worked out a satisfying (to me) rationalisation for that as well, after which I was content to move on.
In the case of Cauchy's Bound, you opined that its proof was a bit beyond the scope of the text. I couldn't rationalise it for myself, so I went looking for a proof, and found an "accessible" statement of proof here:
I've also seen this proof, or very similar versions of it, elsewhere on the 'Net.
In cases like this, I sometimes like to prepare my own explanatory document, which might include more explicit statements of what might otherwise be leaps of assumptions – whatever works best for me.
Cauchy's Bound is stated to produce – using the variable "M" as you use it in Precalculus 3 – a closed interval [–(M + 1), M + 1] within which all real zeros of the polynomial lie. However, as I worked through it myself, the process seemed to me to be producing an OPEN interval (–(M + 1), M + 1) instead. Granted, seeing that a closed interval [a, b] includes the open interval (a, b), there seems to me to be no harm in this case "overstating" an open interval as a closed one. However, I am bothered by having obtained a slightly different conclusion, so, would you mind having a look at what unfolded for me below, please?
... starting to the left of where the heading "Archives" occurs near the right-hand side of the page. The author of that page uses the variable "h" where you (and I) use the variable "M".
Corresponding to that is part of my document:
The (unshown) assumptions stated earlier in the text in my document are that:
* P(x) is a polynomial with real coefficients and of degree n (n >= 1).
* P(x) = 0.
* |x| > 1.
* M > 0 (ie, P(x) is at least a binomial). If M = 0, then P(x) is simply a monomial with x = 0.
At step (13) in my document, I started looking at the value of (1 / |x|^n), reaching the slightly unconventional conclusion at step (18) of "|x| < M + 1" instead of the usual "|x| <= M + 1".
I then looked at the two remaining trivial scenarios concerning P(x) to see if they too would be correctly accommodated by the general conclusion that I reached in step (18). (They appear to be.)
Carl, I appreciate that this is pedantry of a fairly high order, but do you see any errors of derivation through steps (11)-(18) in my document that result in a slightly different conclusion for Cauchy's Bound? I'm wondering if it is conventional to state Cauchy's Bound as a closed interval just to be "safe" and because it doesn't do any practical harm to do so.
Thank you for your fast reply, Carl. Correct me if I'm wrong, but I get two takeaways from this:
1. Statement of a prediction of Cauchy's Bound as a closed interval has become a conventional presentational nicety. Maybe, from a paedagogical perspective, a closed interval sits more intuitively with an audience acquiring this as new knowledge?
2. However, don't bother testing the endpoint values of the closed interval as potential zeros of the polynomial because, strictly speaking, they're outside the predicted bounds.