Is Every Convergent Sequence Also Contractive?

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SUMMARY

The discussion centers on the relationship between convergent sequences and contractive sequences. It is established that while every contractive sequence is convergent, the converse is not true; a convergent sequence may not be contractive. A counterexample provided is the sequence .9, 1, 1, .99, 1, 1, .999, 1, 1, .9999, which converges to 1 but does not exhibit the properties of a contractive sequence due to the presence of constant differences.

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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.
 

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