Can infinitessimals be represented as monotone decreasing sequences?

In summary, the conversation is about trying to prove that any infinitesimal can be written as a monotone decreasing sequence. The person is unsure if it is true and is seeking ideas for a proof. They suggest using a subsequence that is monotone decreasing, but are unsure about how to choose the set of n \in N corresponding to the subsequence in the ultrafilter F. The conversation ends with the idea that if it is not true, it could potentially be used for a refinement of the filter.
  • #1
jem05
56
0
Hello,
Happy holidays everyone,

I'm trying to prove that any infinitessimal can be written as a monotone decreasing sequence; that is, one of its representations as a sequence of real numbers is a mon. dec. seq.
I'm really stuck, and i don't even know if it's true.
Intuitively, it should work.

I mean i can get a subsequence that is monotone decreasing since the infinitessimal is smaller than any real number,
but how do i know this set of n [tex]\in[/tex] N corresponding to the subsequence chosen [tex]\in[/tex] ultrafilter F.

Any ideas?
Thanks.
 
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  • #2
I think if it is not, then it can be used for a refinement of the filter, but there is none by definition of ultrafilter.
 

FAQ: Can infinitessimals be represented as monotone decreasing sequences?

What is non standard analysis?

Non standard analysis is a mathematical framework that extends the traditional real numbers to include infinitesimals and infinitely large numbers. It is based on the work of mathematicians Abraham Robinson and Edwin Hewitt and allows for a more rigorous treatment of infinitesimals in calculus and other areas of mathematics.

How is non standard analysis used in mathematics?

Non standard analysis provides a more intuitive and rigorous way of dealing with infinitesimals and infinitely large numbers. It has applications in areas such as calculus, differential equations, and probability theory. It has also been used in mathematical physics and computer science.

What are some advantages of using non standard analysis?

One advantage of non standard analysis is that it allows for a more natural and intuitive approach to calculus, especially in dealing with limits and infinitesimals. It also provides a rigorous foundation for the use of infinitesimals, which were previously considered to be a "non-rigorous" concept in mathematics.

Are there any limitations to non standard analysis?

One limitation of non standard analysis is that it requires a significant amount of mathematical background and understanding of the framework to use it effectively. It may also be more difficult to apply in certain areas of mathematics, such as algebra and number theory.

Is non standard analysis widely accepted in the mathematical community?

Non standard analysis has gained acceptance in the mathematical community since its development in the 1960s. It is now considered a valid and useful tool in many areas of mathematics. However, there are still some mathematicians who prefer to use traditional methods and do not accept the use of infinitesimals in calculus.

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