Epsilon Argument: Does "a<r+eps" Imply "a<=r"?

In summary, the conversation discusses the concept of showing a<=r and how it can be proven using an argument involving epsilons. The statement a<=r can be implied by showing that for every epsilon greater than 0, a is less than r+eps. It is also noted that if a is greater than r, the argument would be invalid. The conversation ends with gratitude for the assistance provided.
  • #1
magnumartus
4
0
Hi,

Want to show: a<=r

However I ended up with such an argument: For each eps>0, a<r+eps. Does this statement imply a<=r?
 
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  • #2
magnumartus said:
Hi,

Want to show: a<=r

However I ended up with such an argument: For each eps>0, a<r+eps. Does this statement imply a<=r?

No, this implies that you can find any epsilon that when added to r makes it larger than a. This would work even when a is not equal to r, and is lesser. for eg a = 1, r = 2.
 
  • #3
magnumartus said:
Hi,

Want to show: a<=r

However I ended up with such an argument: For each eps>0, a<r+eps. Does this statement imply a<=r?

This does show a <= r. If a>r, then a-r is a positive number. In particular, you then showed that

a<r+(a-r) (picking eps=a-r)

Hence a<a. That doesn't make any sense, so it must have been a<=r the whole time
 
  • #4
hey, it helped a lot. thank you all
 

Related to Epsilon Argument: Does "a<r+eps" Imply "a<=r"?

1. What is the Epsilon Argument?

The Epsilon Argument is a mathematical concept used to show that a given value is either less than or equal to another value. It is commonly used in proofs and arguments related to inequalities.

2. What does "a

"a

3. Does "a

Yes, it does. If a is less than r plus epsilon, then it must also be less than or equal to r. This is because epsilon is a small positive number, so adding it to r will not change its value significantly.

4. Why is the Epsilon Argument useful?

The Epsilon Argument is useful because it allows us to show that a given value is very close to another value, without having to prove that they are exactly equal. This can be helpful in situations where exact values are difficult to determine, but we still want to show a relationship between them.

5. Can the Epsilon Argument only be used with numbers?

No, the Epsilon Argument can be used with any type of mathematical object, as long as there is a defined notion of "closeness" between them. For example, it can be used with functions, matrices, and other types of variables.

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