## Tuesday, February 03, 2009

### An Inequality Lemma for the Cauchy Bound of Real Roots

The following result is used my proof of Cauchy's Bound for real roots.

Lemma 1: abs(c)
n = abs(cn)

Proof:

(1) Assume that c is nonnegative

(2) Then, cn is nonnegative

(3) Then, abs(cn) = cn

(4) Since abs(c) = c, it follows that: cn = abs(c)n

(5) Assume that c is negative

(6) We can assume that n is odd

[Otherwise, cn = (-c)n = abs(c)n = abs(cn) ]

(8) abs(cn) = -cn = (-1)n*cn = (-c)n = abs(c)n

QED

Lemma 2:

Let:

ancn = -an-1cn-1 + .... + -a0

Then:

abs(an)*abs(c)n ≤ abs(an-1)*abs(c)n-1 + ... + abs(a0)

Proof:

(1) Using the Triangle Inequality (see Lemma 4, here), we know that:

abs(-an-1cn-1 + .... + -a0) ≤ abs(-an-1cn-1) + ... + abs(-a0)

so that:

abs(ancn) ≤ abs(-an-1cn-1) + ... + abs(-a0)

(2) Using a basic property of inequalities (see Lemma 1, here):

abs(an)*abs(cn) = abs(ancn)

and likewise:

abs(-an-1)*abs(cn-1) = abs(-an-1cn-1)

...

(3) So we have:

abs(an)*abs(cn) ≤ abs(-an-1)*abs(cn-1) + ... + abs(-a0)

(4) Using Lemma 1 above, we have:

abs(an)*abs(c)n ≤ abs(an-1)*abs(c)n-1 + ... + abs(a0)

QED