Real analysis proof with sequences

In summary, the conversation discusses proving the statement that the limit of a sequence Sn in R is equal to 0 if and only if the limit of the absolute value of Sn is also equal to 0. It is mentioned that this may seem like circular logic, but it is in fact a valid proof. The attempt at a solution involves assuming both limits and using similar arguments to show that they are equivalent.
  • #1
koab1mjr
107
0

Homework Statement


Let Sn be a sequence in R

Prove lim Sn= = 0 if and only if lim abs(Sn) = 0

Homework Equations



none

The Attempt at a Solution



I think this is someone ciruclar logic and that is why I am stuck

Assume lim Sn = 0, thus for n > N implies |Sn| < epsilon or -epsilon < Sn < epsilon. Since Sn and -Sn are less than epsilon |Sn| < Epsilon for sufficently large n.
Now assume lim |Sn|= 0 so ||Sn||< epsilon and using a similar argument show Sn < Epsilon and complete the proof. IS that the way to go?
 
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  • #2
It's not circular, it's just that proving Sn->0 and |Sn|->0 are pretty much the same thing. Yes, that's the way to go.
 

FAQ: Real analysis proof with sequences

1. What is the purpose of using sequences in real analysis proofs?

Sequences are used in real analysis proofs to demonstrate the behavior and properties of functions and sets. They can help prove the convergence or divergence of a function or set, as well as provide insights into continuity, differentiability, and integrability.

2. How are sequences defined in real analysis?

A sequence in real analysis is a function that maps natural numbers to elements of a set. It can be represented as {an}, where n is a natural number and an is the corresponding element in the set. The sequence can be finite or infinite, and the elements in the sequence can follow a specific pattern or be randomly chosen.

3. What is the importance of the Bolzano-Weierstrass theorem in real analysis proofs with sequences?

The Bolzano-Weierstrass theorem states that every bounded sequence in real numbers has a convergent subsequence. This is a fundamental theorem in real analysis, and it is often used in proofs involving sequences to show the existence of a limit or to prove the convergence of a sequence.

4. Can sequences be used to prove the continuity of a function?

Yes, sequences can be used to prove the continuity of a function. In particular, if the limit of a sequence of points in the domain of a function is equal to the value of the function at a particular point, then the function is continuous at that point.

5. Are there any common misconceptions about using sequences in real analysis proofs?

One common misconception is that a sequence always converges to a single value. In reality, a sequence can converge to multiple values or not converge at all. Additionally, the terms in a sequence do not have to be strictly increasing or decreasing, and they can repeat. Another misconception is that proving the convergence of a sequence automatically proves the convergence of a function, which is not always the case.

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