Double sequence convergent iff all of its subsequence converge?

Question

For sequences of a single variable we know that the sequence is convergent if and only if all of its subsequences converge to the same limit as the original sequence. Does this fact hold for double sequences?

Answer 1

The precise statement is $\{a_{nm}\} \to a$ if and only if $\{a_{{n_k}{m_k}}\} \to a$ whenever $\{n_k\}$ and $\{m_k\}$ both increase to $\infty$. 'Only if' part is immediate from definition. For the 'if' part suppose $\{a_{nm}\}$ does not converge to a. Then there exists $\epsilon >0$ and integers $n_k$, $m_k$ such that $| a_{{n_k}{m_k}}-a| \geq \epsilon$ for all k. We may assume that $\{n_k\}$ and $\{m_k\}$ are both increasing. Thus we have produced a subsequence that does not converge.