Proving Inequalities: Three Exercises from Velleman's How to Prove It"

[Government$]

I am doing exercises form Velleman's How to Prove It

Homework Statement

1. Suppose a and b are real numbers. Prove that if a < b < 0 then a^2 > b^2.
2. Suppose a and b are real numbers. Prove that if 0 < a < b then 1/b < 1/a.
3. Suppose a and b are real numbers. Prove that if a < b then (a+b)/2 < b.

2. The attempt at a solution

1. Since a < b < 0 , this means that both a and b are negative numbers. Then i multiply
a < b by a and get a^2 > ab and then again multiply a < b by b and get ab > b^2.
Then since, a^2 > ab and ab > b^2 therefore a^2 > b^2.

2. I multiply a < b by ab and get 1/b < 1/a.

3. I add b to a < b and get a + b < 2b and divide this by 2 and get (a+b)/2 < b

 

[HallsofIvy]

I am doing exercises form Velleman's How to Prove It

Homework Statement

1. Suppose a and b are real numbers. Prove that if a < b < 0 then a^2 > b^2.
2. Suppose a and b are real numbers. Prove that if 0 < a < b then 1/b < 1/a.
3. Suppose a and b are real numbers. Prove that if a < b then (a+b)/2 < b.

2. The attempt at a solution

1. Since a < b < 0 , this means that both a and b are negative numbers. Then i multiply
a < b by a and get a^2 > ab and then again multiply a < b by b and get ab > b^2.
Then since, a^2 > ab and ab > b^2 therefore a^2 > b^2.

Yes, that's excellent.

2. I multiply a < b by ab and get 1/b < 1/a.

Well, you meandivide by ab, not multiply. And you should specifically say that "because 0< a< b, ab> 0".

3. I add b to a < b and get a + b < 2b and divide this by 2 and get (a+b)/2 < b

Yes. Notice that the same kind of argument shows that a< (a+b)/2 so that the "mean" of two distinct numbers always lies between them.

 

[Government$]

Thank you for response.

Yes, that's excellent.


Well, you meandivide by ab, not multiply. And you should specifically say that "because 0< a< b, ab> 0".


Yes. Notice that the same kind of argument shows that a< (a+b)/2 so that the "mean" of two distinct numbers always lies between them.

2. Yes i meant divide not multiply.
3. I haven't noticed that but it's a nice little insight.

So, is the key in proofs to manipulate equation to get for A to B without putting in numbers since that isn't proof?