# Cauchy's Bound: closed or open interval? Classic List Threaded 3 messages Open this post in threaded view
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## Cauchy's Bound: closed or open interval?

 Hello Carl, 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: https://captainblack.wordpress.com/2009/03/08/cauchys-upper-bound-for-the-roots-of-a-polynomial/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? Firstly, I refer you to: https://captainblack.wordpress.com/2009/03/08/cauchys-upper-bound-for-the-roots-of-a-polynomial/... 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.