# convering subsequences

• Sep 21st 2009, 09:10 AM
friday616
convering subsequences
Provide an example of a sequence with a given property:

(a) a sequence that has subsequences that converge to 1, 2, and 3.
(b) a sequence that has subsequences that converge to infinity and negative infinity.
(c) a sequence that has a strictly increasing subsequence, a strictly descreasing subsequence, and a constant subsequence.
(d) an unbounded sequence which has a convergent subsequence
(e) a sequence that has no convergent subsequence

Thanks for any help!
• Sep 21st 2009, 09:29 AM
Plato
Quote:

Originally Posted by friday616
Provide an example of a sequence with a given property:
(a) a sequence that has subsequences that converge to 1, 2, and 3.

I will give you example for this one.
You try the others.
$x_n = \bmod (n,3) + 1 + \frac{1}{n}$
• Sep 21st 2009, 09:05 PM
putnam120
For (e) does $\pm\infty$ count as limits? Because if they do and $\limsup$ and $\liminf$ always exist in the extended reals, the problem is impossible (in $\mathbb{R}$ at least)

If not then use $a_n=n$