Limits of Inequality: Proving by Contradiction

In summary, the statement proves that if f(x) is less than or equal to g(x), then the limit of f(x) is less than or equal to the limit of g(x) as x approaches a, as long as both limits exist. This can be proven by contradiction, by assuming that the limit of f(x) is greater than the limit of g(x) and choosing epsilon to be half the difference of the two limits. This choice of epsilon leads to a contradiction, proving the original statement.
  • #1
Buri
273
0

Homework Statement



If f(x) <= g(x) then lim[x->a] f(x) <= lim[x->a] g(x) provided that both of these limits exist.


2. The attempt at a solution

I've been able to prove it by contradiction. So I assumed that l = lim[x->a] f(x) > lim[x->a] g(x) = m. Therefore, l - m > 0 and I could choose epsilon = (l - m)/2 and the contradiction follows. However, what I need someone to help me with is how can I "see" that (l - m)/2 will actually work (i.e. yield a contradiction)? I just immediately thought of using this choice of epsilon and a contradiction followed. But going back to the problem now, I have no clue why I choose this epsilon. Can someone be able to help me as to why this one actually works?

Thanks
 
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  • #3
Mark44 said:
(l - m)/2 is half the difference of l and m.

LOL! Obviously...
 

Related to Limits of Inequality: Proving by Contradiction

1. What is the concept of "Limits of Inequality: Proving by Contradiction"?

"Limits of Inequality: Proving by Contradiction" is a mathematical concept that involves using a proof by contradiction to establish the limits of inequality between two quantities. This method involves assuming the opposite or contradictory statement and then showing that it leads to a contradiction, thus proving the original statement to be true.

2. How is this concept used in mathematics?

This concept is commonly used in mathematics to prove inequalities, such as those found in calculus or linear algebra. It is often used to establish the upper or lower bounds of a quantity, or to show that a certain quantity is less than or greater than another.

3. Can you provide an example of using "Limits of Inequality: Proving by Contradiction" in a proof?

One example of using this concept in a proof is in proving that the square root of 2 is an irrational number. Assume the opposite, that the square root of 2 is rational, and show that this leads to a contradiction (such as an irrational number being equal to a rational number). This proves that the original statement, that the square root of 2 is irrational, is true.

4. What are the benefits of using this method in proofs?

Using "Limits of Inequality: Proving by Contradiction" can be beneficial in proofs because it allows for a more concise and elegant solution. It also helps to avoid lengthy and complex algebraic manipulations that may be required in other methods of proof.

5. Are there any limitations to using this method in mathematics?

While this method can be effective in proving inequalities, it may not always be suitable for all types of mathematical problems. It also requires the ability to think logically and creatively, which may be challenging for some individuals.

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