I have come-across the following situation in my reading about double sequences $\{s_{m,n}\}_{m,n \in \mathbb{N}}$.
In my situation, it is supposed $\lim_{n\rightarrow \infty} s_{m,n}$ exists as a definite number, for all values of $m$ (note: this is not a hypothesis which necessarily implies the following statement, it is just something which is given and may or may not be important for this question). What is meant by the statement: "the sequences \begin{eqnarray} \{s_{m,1}\}_{m \in \mathbb{N}} \ , \ \{s_{m,2}\}_{m \in \mathbb{N}} \ , \ldots \ , \ \{s_{m,n}\}_{m \in \mathbb{N}} \ , \ \ldots \quad \quad \quad \quad (1)\end{eqnarray} are uniformly convergent with respect to $m$"?
I know of the definition of uniform convergence for sequences of functions $\{f_{m}(x)\}_{m\in \mathbb{N}}$ over some interval $(a,b)$. But the context of (1) is that $s_{m,n}$ are just sequences of numbers, not functions, i.e. there's no variable "$x$".
Thank you for answers and comments