Show-that-a-b-R-1-x-y-x-y-2-1-xy-1-1-x-1-1-y-1-

Question Number 164622 by mathocean1 last updated on 19/Jan/22

S h o w t h a t \forall a, b \in R,

1 . ∣∣ x ∣ - ∣ y ∣∣⩽∣ x - y ∣

2 . 1 + ∣ x y - 1 ∣⩽ (1 + ∣ x - 1 ∣) (1 + ∣ y - 1 ∣) .

Answered by hmrsh last updated on 19/Jan/22

Answer of part (1) :

* ∣ x ∣ ⩽ A \leftrightarrow - A ⩽ x ⩽ A

* * ∣ x + y ∣ ⩽ ∣ x ∣ + ∣ y ∣ (triangle inequality)

∣ x ∣ = ∣ (x - y) + y ∣

You can't use 'macro parameter character #' in math mode

also :

∣ y ∣ = ∣ (y - x) + x ∣ ⩽ ∣ y - x ∣ + ∣ x ∣=∣ x - y ∣ + ∣ x ∣

\to ∣ y ∣ - ∣ x - y ∣ ⩽ ∣ x ∣

You can't use 'macro parameter character #' in math mode

\to - ∣ x - y ∣ ⩽ ∣ x ∣ - ∣ y ∣ ⩽ ∣ x - y ∣

\overset{*}{\to} ∣∣ x ∣ - ∣ y ∣∣ ⩽ ∣ x - y ∣

Commented by mathocean1 last updated on 19/Jan/22

t h a n k s .

Answered by hmrsh last updated on 20/Jan/22

Answer of part 2 :

tip 1 :

x y - 1 = (1 - x) (1 - y) + (x - 1) + (y - 1)

tip 2 :

(1 + x_{1}) (1 + x_{2}) = 1 + x_{1} + x_{2} + x_{1} x_{2}

using triangle inequality and tip 1 :

∣ x y - 1 ∣ = ∣ (1 - x) (1 - y) + (x - 1) + (y - 1) ∣

⩽ ∣ (1 - x) (1 - y) ∣ + ∣ (x - 1) + (y - 1) ∣

⩽ ∣ (1 - x) (1 - y) ∣ + ∣ x - 1 ∣ + ∣ y - 1 ∣

= ∣ 1 - x ∣∣ 1 - y ∣ + ∣ x - 1 ∣ + ∣ y - 1 ∣

= ∣ x - 1 ∣∣ y - 1 ∣ + ∣ x - 1 ∣ + ∣ y - 1 ∣

\to ∣ x y - 1 ∣⩽∣ x - 1 ∣∣ y - 1 ∣ + ∣ x - 1 ∣ + ∣ y - 1 ∣

\overset{+ 1}{\to} ∣ x y - 1 ∣ + 1 ⩽∣ x - 1 ∣∣ y - 1 ∣ + ∣ x - 1 ∣ + ∣ y - 1 ∣ + 1

using tip 2 to RHS :

∣ x y - 1 ∣ + 1 ⩽ (1 + ∣ x - 1 ∣) (1 + ∣ y - 1 ∣)

Commented by mathocean1 last updated on 22/Jan/22

t h a n k s s i r .