If a sequence $a_n$ has exactly three partial limits, and a sequence $b_n$ has exactly two partial limits. could the sequence $c_n=a_n+b_n$ converge? If a sequence $a_n$ has exactly three partial limits, and a sequence $b_n$ has exactly two partial limits. is it possible for the sequence $c_n=a_n+b_n$ to converge? 
I think this question doesn't demand a proof for every two sequences that have the same features, i think there is some examples for sequences that work and some that don't. for now I could think of examples for sequences that only disprove this allotment.
For example $a_n= 1,2,3,1,2,3,1,2,3,...$
and $b_n=-1,1,-1,1,-1,1,-1,..$
then $c_n=0,3,2,4,1,4,0,3,2...$ 
do you gues have an example for sequences that might work? 

Suppose $(a_n+b_n)$ converges to $L$. Suppose further that $(a_n)$ has a subsequence that converges to $M$. Then, since any subsequence of $(a_n+b_n)$ converges to $L$, it follows that $(b_n)$ has a subsequence (corresponding to the subsequence of $(a_n)$ mentioned above) converging to $L-M$.
The same holds with the roles of $(a_n)$ and $(b_n)$ interchanged. From this, it follows that the set of limit points of $(a_n)$ and the set of limit points of $(b_n)$ have the same cardinality.
In particular, the situation you describe can not occur.
