~~簡易不等式(3) <--> 絕對值不等式下篇~~

上篇已證 :
| |a| - |b| | ≤ |a + b| ≤ |a| + |b| is true for all a,b are real number.
Replace b to (-b)
| |a| - |-b| | ≤ |a + (-b) | ≤ |a| + |-b| is true for all a,b are real number.
Hence, | |a| - |b| | ≤ |a - b| ≤ |a| + |b| for any a,b are real number.

2011-06-22 21:33:16 補充：
by definition, a ≤ |a| and b ≤ |b| for any real number
| a+b | ≤ | |a| + |b| | = |a| + |b| (since |a|, |b|>0 )
'' |a + b| ≤ |a| + |b| '' ---(1) is true for all a,b are real number
Using (1) with a|-->a+b and b|--> -b
|(a + b)+(-b)| ≤ |a+b| + |-b|
|a| ≤ |a+b| +|b| (since |b|=|-b|)
=>|a| - |b| ≤ |a + b| ------(2)
Since a,b are real, by (2)
=> |a| - |b| >= -|a + b| (since |a+b|>0)-----(3)
by (2),(3), - |a + b| ≤ |a| - |b| ≤ |a + b|
i.e. | |a| - |b| | ≤ |a + b| ------(4)
by (1),(4) ,
| |a| - |b| | ≤ |a + b| ≤ |a| + |b| are true for all a,b are real number
Now, Replace b to (-b)
| |a| - |-b| | ≤ |a + (-b) | ≤ |a| + |-b| is true for all a,b are real number.
Hence, | |a| - |b| | ≤ |a - b| ≤ |a| + |b| for any a,b are real number.

Thankyou!!