How Many Subsequential Limits Does the Sum of two Bounded Sequences Have? I ask for some help or hint how to deal with this question:

Suppose $a_n$,$b_n$ were bounded sequences, 
  $a_n$ has $k$ sub-sequential limits and $b_n$ has $m$ sub-sequential limits.
Prove or provide a counterexample :
  $a_n$+$b_n$ has at most $km$ sub-sequential limits.

I didn't find any counterexample, but 
I didn't find a right approach to prove this statement also.
Probably I didn't catch the meaning of the sub-sequential limits concept,
so any links to right direction will be welcome.
Thanks.
 A: First prove that if $\langle a_n:n\in\Bbb N\rangle$ is a bounded sequence with $k$ subsequential limits $x_1,\ldots,x_k$, then $\Bbb N$ can be partitioned into $k$ infinite sets $A_1,\ldots,A_k$ such that $\langle a_n:n\in A_i\rangle\to x_i$ for $i=1,\ldots k$. To do this, take a subsequence converging to $x_1$; from the remaining terms take a subsequence converging to $x_2$, and so on. Show that these $k$ subsequences use up all but at most finitely many terms of the original sequence, and those can simply be thrown in with the first subsequence.
Now let $y_1,\ldots,y_m$ be the subsequential limits of $\langle b_n:n\in\Bbb N\rangle$; by the result of the previous paragraph we can partition $\Bbb N$ into $m$ infinite sets $B_1\ldots,B_m$ such that $\langle b_n:n\in B_i\rangle\to y_i$ for $i=1,\ldots,m$. Suppose that $i\in\{1,\ldots,k\}$, $j\in\{1,\ldots,m\}$, and $A_i\cap B_j$ is infinite; then $\langle a_n+b_n:n\in A_i\cap B_j\rangle$ is an infinite subsequence of $\langle a_n+b_n:n\in\Bbb N\rangle$ converging to $x_i+y_j$. (Why?) Thus, $\langle a_n+b_n:n\in\Bbb N\rangle$ can have at least $km$ subsequential limits: we need only arrange matters so that the $km$ sums $x_i+y_j$ are all distinct and the intersections $A_i\cap B_j$ are all infinite. I leave it to you to show that this can be done for any $k$ and $m$.
Now suppose that $C\subseteq\Bbb N$ is infinite and that the subsequence $\langle a_n+b_n:n\in C\rangle$ converges to some $z\in\Bbb R$. We’d like to show that $z=x_i+y_j$ for some $i\in\{1,\ldots,k\}$ and $j\in\{1,\ldots,m\}$.


*

*There must be an $i\in\{1,\ldots,k\}$ and a $j\in\{1,\ldots,m\}$ such that $C\cap A_i\cap B_j$ is infinite; why?  

*What can you say about the subsequence $\langle a_n+b_n:n\in C\cap A_i\cap B_j\rangle$?

