Double sequence

Remarks

A double sequence is a sequence indexed by $\N \times \N$ instead of $\N$. Example: $x_{m,n} = m/(m + n + 1)$ is a double sequence of real numbers. For fixed $m$ it tends to $0$ as $n \to \infty$, and for fixed $n$ it tends to $1$ as $m \to \infty$. So $\lim_m \lim_n x_{m,n} = 0$ and $\lim_n \lim_m x_{m,n} = 1$: the two iterated limits differ. Non-example: a triangular array $(x_{m,n})_{0 \le n \le m}$ is not a double sequence, since it is defined only for $n \le m$; it becomes one after extension by $0$. The joint limit in the sense of Pringsheim convergence is the usual notion of limit for double sequences.