In Analysis I Tao states this lemma: Let ε, δ > 0. If x and y are ε-close, and z and w are δ-close, then xz and yw are (ε|z| + δ|x| + εδ)-close.
Here x and y being ε-close is defined by |x - y| ≤ ε. He proofs it like this:
Let ε, δ > 0, and suppose that x and y are ε-close. If we write a := y − x, then we have y = x + a and that |a| ≤ ε. Similarly, if z and w are δ-close, and we define b := w − z, then w = z + b and |b| ≤ δ.
Since y = x + a and w = z + b, we have yw = (x + a)(z + b) = xz + az + xb + ab.
Thus |yw − xz| =
= |az + bx + ab| ≤
≤ |az| + |bx| + |ab| =
= |a| |z| + |b| |x| + |a| |b|.
Since |a| ≤ ε and |b| ≤ δ, we thus have |yw − xz| ≤ ε|z| + δ|x| + εδ and thus that yw and xz are (ε|z| + δ|x| + εδ)-close. ∎
Somewhere I else i found this proof for an unstated lemma
yw - xz =
= yw - xw + xw - xz =
= y(w - z) + z(y - x)
And so
|yw - xz| =
= |y(w - z) + z(y - x)| ≤
≤ |y(w - z)| + |z(y - x)| =
= |y| |w - z| + |z| |y - x|
= |y| |b| + |z| |a| ≤
≤ δ |y| + ε |z| ∎
To restate the unknown lemma in the Tao-way: Let ε, δ > 0. If x and y are ε-close, and z and w are δ-close, then xz and yw are (ε|z| + δ|y|)-close.
So |x| got substituted by |y| and the product εδ vanished. The main difference between them is that x and z are factors in one term of the difference (here the subtrahend) , while y and z are in different terms. Does this property alone give the relaxation of the εδ-term? Or was it superfluous from the start?
In the absolute value |x - y| = |y - x| so i can always switch x and y. This makes me wonder a bit.
As thinking seemed so hard that i had to write an stackexchange question. I tried to not think and just made up a simple example, which i can look at without any wheres or whys.
Let x = 2, y = 3, w = 2 and z = 1 here ε = |2 − 1| = 1 = |3 − 2| = δ
Then p = yw − xz = 3·2 − 2·1 = 6 − 2 = 4
the formula with absolute values from factors of different terms (also from different pairs for which closeness was stated) ε|y| + δ|z] = 1·3 + 1·1 = 4 ≥ p ε|z| + δ|w] = 1·2 + 1·2 = 4 ≥ p
Then Tao's formula with the absolute values from factors of the same term (which then are automatically from different pairs for which closeness was defined. ε|x| + δ|z] + εδ = 1·2 + 1·1 + 1·1 = 4 ≥ p ε|y| + δ|w] + εδ = 3·1 + 2·1 + 1·1 = 6 ≥ p
For Taos formula without the εδ-term: ε|x| + δ|z] + εδ = 1·2 + 1·1 = 3 < p
So if i choose the absolute values from factors of the same term i need the εδ-term.
To the where s and whys, which i thought about during writing this example: It seems like it has to do with cases. In case that in the products the big partner is multiplied with the small partner of the other pair and vice versa, there is no problem anyways.
Problems only arise if the small partner is teamed up with the other pairs small partner, and in the other term i have the product of the big partner with the other big partner. In that case i can get bot the small partners when i take the factors from the same term, and so underestimate the difference, but i can only get a small and a big partner if i take the absolute values from factors of the different term.
My musings are so colloquial, i will wait a while till accepting them (or rewrite them more mathematically)