Explanation of Cauchy Proof for Convergent Sequence

In summary, the conversation discusses the convergence of a sequence {a_n} to a limit A and the use of a chosen value e to show that for all n, m greater than or equal to a positive integer N, the values a_n and a_m fall within the interval (A-e, A+e). It is then mentioned that the length of this interval is 2e, which is the difference between (A+e) and (A-e).
  • #1
lion0001
21
0
Suppose { a_n } converges to A. Choose e > 0, THere is a positive integer N such that, if
n, m >= N , then A - e < a_n < A + e and A - e < a_m < A + e
Thus for all n, m >= N we find a_n ∈ ( A - e , A + e ) and
a_m ∈ ( A - e , A +e ) . the set ( A - e, A +e ) is an interval of length 2e , hence the difference between a_n, and a_m is less then 2e

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i don't understand how they get 2e , ( a set of length 2e ?? )
 
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  • #2
what is (A + e) - (A - e)
 
  • #3
It's simply (A + e) - (A - e) = 2e.
Same as I would say the interval (2,5) has length 5-2=3.
 
  • #4
nicksauce said:
It's simply (A + e) - (A - e) = 2e.
Same as I would say the interval (2,5) has length 5-2=3.

Excellent answer, OMG this proved that i am an idiot =(
 

FAQ: Explanation of Cauchy Proof for Convergent Sequence

What is a Cauchy proof?

A Cauchy proof is a type of mathematical proof that involves using the Cauchy criterion, which states that a sequence converges if and only if it satisfies certain conditions. These conditions involve the difference between consecutive terms in the sequence becoming smaller and smaller as the sequence progresses.

Why is the Cauchy criterion important?

The Cauchy criterion is important because it allows us to determine whether or not a sequence converges without having to explicitly find its limit. This can be useful in situations where finding the limit is difficult or impossible.

How do I use the Cauchy criterion in a proof?

To use the Cauchy criterion in a proof, you must show that the difference between consecutive terms in the sequence becomes smaller and smaller as the sequence progresses. This can be done by using mathematical induction or other proof techniques.

Can the Cauchy criterion be used for all sequences?

No, the Cauchy criterion only applies to certain types of sequences, specifically those that are bounded and monotonic. If a sequence does not meet these criteria, the Cauchy criterion cannot be used to determine convergence.

Are there any other criteria for determining convergence of a sequence?

Yes, there are other criteria such as the Monotone Convergence Theorem and the Bolzano-Weierstrass Theorem. These criteria may be more applicable to certain types of sequences, so it is important to understand and use the appropriate criterion in a proof.

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