Prove that Schwartz's inequality?

Question

Prove that , if a and b are real numbers :
││a│-│b││≦│a-b│≦│a│+│b│

Answer 1

I will not do this for you -- but will provide some direction.

Break it into five cases:

1. What happens if both A and B are positive

2. What happens if both A and B are negative.

3. What happens if A is positive & B is negative.

4. What happens if A is negative and B is positive.

5. What happens if at least one of A or B is zero
Under each of these cases, you can get rid of some of the absolute value signs & see what happens. For example, in case one, the left hand portion and the middle portion will have the same value and the right side will obviously be greater.

Under case three, the middle and right hand sides will be equal and the left hand side will obviously be lower.