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I am asked to find a sequence and a number such that $$|a_{n+1}-a| \lt |a_n-a|$$, but $a_n$ does not converge to a. Please help!

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up vote 7 down vote accepted

$a_n=1+\dfrac1n$ and $a=0{}{}{}$.

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Or any positive strictly decreasing sequence which does not converges to $0$. – Gastón Burrull Jan 19 '14 at 2:07

Take any monotonically increasing (decreasing) sequence which has a limit $l$ and then pick number greater than (smaller than) $l$ for $a$.

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