Suppose $a_1, a_2, \dots$ is a sequence of convergent sequences (i.e. the $n$th sequence is $a_{n, 1}, a_{n, 2}, \dots$), and set $b_i = \lim{a_i}$ for each $i$. Suppose $\lim b_i$ exists and is equal to $b$. Then is there a subsequence of terms $c_n = a_{i_n, j_n}$ that converge to $b$?
If the result as stated is true, perhaps some restrictions can be made on the choice of $c_n$, such as requiring the picked out $a_{i, j}$ to be chosen in dictionary order (i.e. that $n < m$ implies either $i_n < i_m$ or $i_n = i_m$ and $j_n \leq j_m$.