Prove the following:
 if and only if .
 .
 .
 .
 .
 .
 If then .
 .
 .
 .

Proof. If then .
If , then by definition of 
Proof. If , then and .
If , then and (since 
Proof. If , then , so and .
If , then , so and .
If , then so 
Proof. First, . So, if then .
If then  Proof. Since is defined to be the unique nonnegative square root of , from part (d) we have , and is nonnegative by definition; hence,

Proof. If , then , so Further, , so .
If and , then so . Then, and , so
If and , then so . Then and , so
Finally, if and , then so . Then and , so 
Proof. Using the definition of we have
by part (f). But then since , we have . So,

Proof. Using the triangle inequality and the fact that ,

Proof. Here we use the trick of adding and subtracting , and then the triangle inequality,

Proof. Using a similar trick to part (i), we have
Then, combining this inequality with the one in part (i) and applying the definition of the absolute value,
hmm good….
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