Is the converse of this theorem true or not?

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In summary, the theorem states that if the limit of a sequence of real numbers is zero, then the limit of the sequence of absolute values is also zero.
  • #1
Byeonggon Lee
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Homework Statement
Is converse of this theorem true or not?
Relevant Equations
$$\lim_{n\to\infty} a_n$$
This theorem is from the stewart calculus book 11.1.6
If $$ \lim_{n\to\infty} |a_n| = 0$$, then $$\lim_{n\to\infty} a_n = 0$$
I wonder whether converse of this theorem true or not
 
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  • #2
I would say yes.
In fact, that looks more like a definition for "limit" as it applies to complex numbers than it does as a theorem.
 
  • #3
Byeonggon Lee said:
Problem Statement: Is inverse of this theorem true or not?
Relevant Equations: $$\lim_{n\to\infty} a_n$$

This theorem is from the stewart calculus book 11.1.6
If $$ \lim_{n\to\infty} |a_n| = 0$$, then $$\lim_{n\to\infty} a_n = 0$$
I wonder whether inverse of this theorem true or not

I think "converse" is the correct terminology.

Can you prove it?

What if the limit is non zero?
 
  • #4
I think "converse" is the correct terminology.
Yes you're right. I edited the post.
converse of the theorem:
$$ \lim_{n\to\infty} a_n = 0 \Longrightarrow \lim_{n\to\infty} |a_n| = 0$$
I tried to prove it by using contrapositive
contrapositive of the converse:
$$ \lim_{n\to\infty} a_n \neq 0 \Longrightarrow \lim_{n\to\infty} |a_n| \neq 0$$
I think if $$ \lim_{n\to\infty} a_n = c$$ $$ (c \neq 0) $$
this will be also true by limit laws for sequences:
$$ \lim_{n\to\infty} |a_n| = |c|$$ $$ (c \neq 0) $$
But I don't know what to do when
$$ \lim_{n\to\infty} a_n = \infty $$
 
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  • #5
Byeonggon Lee said:
Yes you're right. I edited the post.
converse of the theorem:
$$ \lim_{n\to\infty} a_n = 0 \Longrightarrow \lim_{n\to\infty} |a_n| = 0$$
I tried to prove it by using contrapositive
contrapositive of the converse:
$$ \lim_{n\to\infty} a_n \neq 0 \Longrightarrow \lim_{n\to\infty} |a_n| \neq 0$$
I think if $$ \lim_{n\to\infty} a_n = c$$ $$ (c \neq 0) $$
this will be also true by limit laws for sequences:
$$ \lim_{n\to\infty} |a_n| = c$$ $$ (c \neq 0) $$
But I don't know what to do when
$$ \lim_{n\to\infty} a_n = \infty $$
You need to be careful. First, the negation of the statement that ##a_n## converges to ##0## is that it does not converge to ##0##. This includes cases where the limit does not exist.

What you have for the contrapositive of the converse is not right, therefore.

But, in any case, you should be able to prove directly that the convergence of ##a_n## to ##0## and the convergence of ##|a_n|## to ##0## are equivalent. Just use the definition of the limit.

You also need to be more careful if the limit is nonzero and should look at the problem carefully.

Note that in this case the convergence of the sequence and the convergence of the absolute value are not equivalent.

The same goes for sequences that diverge to ##\pm \infty##.
 
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  • #6
If ##|a_n - 0| < \epsilon##, then also ##||a_n |-0|<\epsilon##, and the other way around. It's just about taking absolute value one or two times.
 
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  • #7
Trivially true using epsilon- definition. Use that ##||x||=|x|##.
 
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  • #8
Thanks. Definitely better to use the definition than contrapositive.
 
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  • #9
the fact that the absolute value is continuous implies the converse is true. More interesting is the fact that the theorem itself is not always true. I.e. the absolute values lie in the real numbers which is a complete field, but the original points may lie in a non complete space, hence the absolute values may converge in the reals, without the original points themselves converging in the original space. E.g. let the original space be the rationals. Then one can have the absolute values of a sequence of rationals converging in the reals to an irrational number, whence the original points do not converge in the rationals. In general absolute convergence of a sequence of points of a normed vector space implies convergence of the sequence itself if and only if cauchy convergence implies convergence, i.e. completeness holds. I hope this is correct; it has been over 50 years since I learned this from Lynn Loomis, but it is pretty firm in memory.

a reference is Advanced Calculus by Loomis and Sternberg, Theorem 7.11 page 221, and exercise 7.19, p. 223.
 
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Related to Is the converse of this theorem true or not?

What is the converse of a theorem?

The converse of a theorem is a statement that is formed by switching the hypothesis and conclusion of the original theorem. For example, if the original theorem states "If A, then B," the converse would state "If B, then A."

How do you determine if the converse of a theorem is true?

To determine if the converse of a theorem is true, you must prove it using the same logical steps and evidence as the original theorem. If the proof is valid, then the converse is considered to be true.

What if the converse of a theorem is false?

If the converse of a theorem is false, it means that the original theorem is not necessarily true. In other words, the hypothesis and conclusion do not have a cause-and-effect relationship.

Can a theorem and its converse both be true?

Yes, it is possible for a theorem and its converse to both be true. This means that the hypothesis and conclusion have a bidirectional relationship, and either one can imply the other.

Why is it important to consider the converse of a theorem?

Considering the converse of a theorem allows us to fully understand the implications of the original statement. It also helps us to identify any potential errors or exceptions in the original theorem and to strengthen our understanding of the concept.

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