
Real Numbers 1-11
x x y y
x y y
x y x y
= − +
≤ − +
⇒ − ≤ −
(by Triangle Inequality)
, which is thhe required result.
m.
x y x y x y+ = + ⋅ ≥iff 0
Proof directly follows from the definition.
n.
x y x y x y− = − ⋅ ≥iff 0
Proof directly follows from the definition
o.
1
[ ] max{ , }x y x y x y+ + − =
Proof:
Consider
x y x y x y x y x y e
x y x
+ + − = + + − − −
= + + −
max{( ), ( )} ( )
max{ (
by property
yy x y x y
x y x y x y x y
x y
x y
), ( )}
max{ , }
max{ , }
max{ ,
+ − −
= + + − + − +
=
=
2 2
2
}}
max{ , }Hence, x y x y x y+ + − = 2
And this leads to the desired result,
1
x y x y x y+ + −
= max{ , }
Similarly, we can prove the following result.
1
x y x y x