Convergence = "closer and closer"

Question

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!

score 7 · Accepted Answer · answered Jan 19 '14 at 01:54

7

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

answered Jan 19 '14 at 01:54

1

Or any positive strictly decreasing sequence which does not converges to $0$. – Gaston Burrull Jan 19 '14 at 02:07

score 0 · Answer 2 · answered Jan 19 '14 at 02:11

0

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

answered Jan 19 '14 at 02:11

Timotej

2 Answers2