Convergence Proof: |a_n| Converges to 0 if a_n Converges to 0

In summary, a convergence proof is a mathematical proof that demonstrates the behavior of a sequence as it approaches a specific limit or value. It is important because it provides a rigorous and logical basis for understanding the behavior of a sequence and making predictions about its limit. Common techniques used in a convergence proof include the squeeze theorem, the monotone convergence theorem, and the Cauchy criterion. To determine if a sequence is convergent, one can use convergence tests such as the ratio test or the root test. A sequence can only have one limit, but it may have a limit at positive or negative infinity.
  • #1
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Hi,

Here's another question from my analsysi HW. I get that the two sequences are equal but I'm not sure how to write it out. Any help would be great.
Thanks.

Homework Statement



Prove that a sequence [itex]{a_n}[/itex] converges to 0 iff the sequence [itex]{\lvert a_n\rvert}[/itex] converges to 0.

Homework Equations





The Attempt at a Solution

 
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  • #2
If [itex]a_n \rightarrow 0 [/itex] you know that after some n [itex]|a_n|\leq \epsilon/2[/itex]

So what can you tell?

Please show your attempt. [This is as close to trivial as it can be]
 
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FAQ: Convergence Proof: |a_n| Converges to 0 if a_n Converges to 0

What is a convergence proof?

A convergence proof is a mathematical proof that shows that a sequence of numbers or functions converges to a specific limit or value. It is used to demonstrate the behavior of a sequence as its terms approach infinity.

Why is a convergence proof important?

A convergence proof is important because it provides a rigorous and logical basis for understanding the behavior of a sequence. It also allows us to make predictions about the behavior of a sequence and to determine whether it will approach a specific limit or not.

What are some common techniques used in a convergence proof?

Some common techniques used in a convergence proof include the squeeze theorem, the monotone convergence theorem, and the Cauchy criterion. These techniques help to establish the existence and value of a limit for a given sequence.

How can I determine if a sequence is convergent or not?

To determine if a sequence is convergent, you can use one of the techniques mentioned above or other convergence tests such as the ratio test or the root test. These tests involve examining the behavior of the sequence's terms and comparing them to known convergent or divergent sequences.

Can a sequence have multiple limits?

No, a sequence can only have one limit. If a sequence has multiple possible limits, it is considered divergent. However, some sequences may have a limit at positive infinity or negative infinity, which are considered to be additional types of limits.

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