Is Every Convergent Sequence Also Contractive?

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In summary, a contractive sequence is always convergent, but the converse is not necessarily true. A counterexample is a sequence that converges but does not have decreasing successive differences, such as 0.9, 1, 1, 0.99, 1, 1, 0.999, 1, 1, 0.9999, etc.
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Guthrie
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Just a quick question regarding contractive sequences and convergence.

I understand that a contractive sequence is always convergent, but is the converse also true? i.e. If a sequence is convergent then its contractive.

I can't think of a logical proof to this, yet a plausible counterexample escapes me.

I would appreciate any advice to point me in the right direction.

Thank you
 
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I had to look up the definition of contractive sequence. If I got it right, here is a counterexample that you might want.
.9, 1, 1, .99, 1, 1, .999, 1, 1, .9999, etc. This converges to 1, but successive differences never form a decreasing sequence, since every third difference is 0, while the others aren't.
 

FAQ: Is Every Convergent Sequence Also Contractive?

What is a contractive sequence?

A contractive sequence is a sequence of numbers in which each subsequent number is closer to a fixed point than the previous number. In other words, the numbers in the sequence get progressively smaller as the sequence continues.

How is a contractive sequence different from a convergent sequence?

A contractive sequence always decreases in value, while a convergent sequence may increase or decrease before ultimately approaching a fixed point. Additionally, a contractive sequence is always bounded, meaning there is a maximum and minimum value for the sequence, while a convergent sequence may not be bounded.

What are some practical applications of contractive sequences?

Contractive sequences have many practical applications in fields such as economics, physics, and biology. For example, in economics, contractive sequences are used to model the behavior of financial markets and the movement of stock prices. In physics, they are used to describe the behavior of systems that approach a stable equilibrium. In biology, contractive sequences can be used to model the growth and decay of populations.

How is the fixed point of a contractive sequence determined?

The fixed point of a contractive sequence can be found by solving the equation x = f(x), where f(x) is the function that generates the sequence. This fixed point is also known as the attractor of the sequence, as all values in the sequence will eventually approach this point.

Are there any limitations to using contractive sequences?

One limitation of using contractive sequences is that they can only be applied to functions that have a fixed point. Additionally, the rate of convergence of a contractive sequence may vary and may not always converge quickly, making it unsuitable for certain applications. Furthermore, the fixed point of a contractive sequence may be sensitive to initial conditions, meaning small changes in the starting value can result in significantly different outcomes.

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