# Proving a^2 or b^2 is Less than or Equal to n

In summary, if n=ab, then either a^2<=n or b^2<=n, based on the trichotomy of a and b where they can be equal, or one is greater than the other. This is shown by multiplying the inequality by the opposite variable to show that the resulting value is smaller than n.
Suppose n=ab, show that a^2<=n or b^2<=n.

Start with trichotomy: a < b, a = b, or a > b. It's pretty simple.

I'm fairly new to proofs, so can you please check if this is correct.

if a=b, n=a^2 or b^2=n
if a>b, a^2>ab, a^2>n
if b>a, b^2>ab, b^2>n

What you write is correct, though for the last two you should really use the opposite variable (and reverse signs) because you're trying to show that something's smaller, not bigger.

Try multiplying a > b by b instead of by a.

so it should be like this then:

if a>b, ba>b^2, n>b^2
if b>a, ab>a^2, n>a^2

## 1. How do you prove that a^2 or b^2 is less than or equal to n?

To prove that a^2 or b^2 is less than or equal to n, you can use a proof by contradiction. Assume that a^2 or b^2 is greater than n, then you can show that this leads to a contradiction. Therefore, a^2 or b^2 must be less than or equal to n.

## 2. What are the key steps in proving a^2 or b^2 is less than or equal to n?

The key steps in proving a^2 or b^2 is less than or equal to n are:

1. State the proof by contradiction.
2. Assume that a^2 or b^2 is greater than n.
4. Conclude that a^2 or b^2 must be less than or equal to n.

## 3. Can you use mathematical induction to prove a^2 or b^2 is less than or equal to n?

No, mathematical induction is not suitable for proving a^2 or b^2 is less than or equal to n. Mathematical induction is used for proving statements that are true for all natural numbers, while proving a^2 or b^2 is less than or equal to n requires a proof by contradiction.

## 4. How can you apply this concept in real-life situations?

This concept can be applied in various real-life situations, such as in engineering and physics. For example, when designing a bridge, it is important to ensure that the weight or force applied on any part of the bridge does not exceed the maximum weight or force that the bridge can support. This can be represented as a^2 or b^2 being less than or equal to n, where a and b are the dimensions of the bridge and n is the maximum weight or force it can handle.

## 5. Are there any exceptions to this concept?

Yes, there are some exceptions to this concept, such as when dealing with complex numbers. In certain cases, the inequality a^2 or b^2 is less than or equal to n may not hold true. Additionally, this concept may not be applicable in situations where the variables are continuously changing, such as in calculus.

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