# Thread: convering subsequences

1. ## 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!

2. 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.
$\displaystyle x_n = \bmod (n,3) + 1 + \frac{1}{n}$

3. For (e) does $\displaystyle \pm\infty$ count as limits? Because if they do and $\displaystyle \limsup$ and $\displaystyle \liminf$ always exist in the extended reals, the problem is impossible (in $\displaystyle \mathbb{R}$ at least)

If not then use $\displaystyle a_n=n$