Is Every Convergent Sequence Also Contractive?

[Guthrie]

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

 

[mathman]

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.