# Analysis Help; proofs via axioms

• silvermane
In summary, the problem is to prove that for any real numbers a, b, c, the expression (a+b+c)^2 is less than or equal to 3 times the sum of the squares of a, b, and c. The permitted axioms to use include properties of inequalities and multiplication with real numbers. A helpful approach is to simplify the problem and work backwards, using the properties to justify each step.
silvermane
Gold Member
Analysis Help; proofs via axioms :)

1. The problem statement:

Prove that for any real numbers a, b, c,
$$(a+b+c)^2\leq3*(a^2 +b^2+c^2)$$​

2. These are the axioms we are permitted to use:

01) Exactly one of these hold: a<b, a=b, or b<a
02) If a<b, and b<c, then a<c
03) If a<b, then a+c < b+c for every c
04) If a<b and 0<c, then ac<bc.

## The Attempt at a Solution

It follows from problem number 1, that
$$0\leq(a + b)^2$$

So can we say that
$$0\leq(a + b + c)^2$$
and would this help me to solve the proof? I'm having a great deal of trouble getting used to thinking like this; I'm very used to thinking combinatorially.
I do know that this is a special case of the Cauchy-Schwartz Sequence where we have
$$(a_{1}+a_{2}+a_{3})^2\leq3*(a_{1}^2 +a_{2}^2+a_{3}^2)$$​
I'm just not allowed to use that fact to proove it

Any help/hints would be greatly appreciated, and I always do my best to return the favor! Thank you so much for your time!

A good way to work on a problem like this is to simplify it the only ways you know how and then work backwards, maybe modifying your work, to justify your steps. Expand the left hand side. You will get three squares and a multiple of 2. Now use your property (3) to cancel out the three squares and consider property (4). I hope things have cleared up after this. After you've seen the way you can work backwards to justify your steps.

VeeEight said:
A good way to work on a problem like this is to simplify it the only ways you know how and then work backwards, maybe modifying your work, to justify your steps. Expand the left hand side. You will get three squares and a multiple of 2. Now use your property (3) to cancel out the three squares and consider property (4). I hope things have cleared up after this. After you've seen the way you can work backwards to justify your steps.

Haha yes, this is more than helpful. Thank you for this hint, it will be well used, and is well appreciated.

Great! Glad you have worked it out

## 1. What are axioms in analysis?

Axioms in analysis are fundamental statements or assumptions that are accepted as true without being proved. They serve as the starting point for all mathematical proofs and help to establish the basic rules of the system being studied.

## 2. How are axioms used in mathematical proofs?

Axioms are used as the building blocks for mathematical proofs. By starting with a set of accepted axioms, mathematicians use logical reasoning to derive new theorems and statements.

## 3. What is the importance of axioms in analysis?

Axioms are essential in analysis as they provide the foundation for all mathematical proofs and help to establish the rules and principles of the system being studied. They also ensure that mathematical arguments are consistent and valid.

## 4. Can axioms be changed or modified?

In most cases, axioms are considered to be unchangeable and are accepted as true without being questioned. However, in certain cases, mathematicians may propose new axioms or modify existing ones in order to explore new mathematical systems or theories.

## 5. How do axioms differ from definitions in analysis?

Axioms and definitions serve different purposes in analysis. Axioms are fundamental assumptions that are accepted as true, while definitions are used to clarify the meaning of mathematical terms. Axioms are often used to prove the validity of definitions.

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