$S(n,m)$ is a double sequence. Can anyone give me an example where lim$_{m , n \to \infty} S(n,m)$ exists but lim$_{n \to \infty}$( lim$_{m \to \infty} S(n,m)$) , lim$_{m \to \infty}$( lim$_{n \to \infty} S(n,m)$) do not? My Attempt: I thought of an example. $S(1 ,m) =m $, $S(n , 1) ...

What is the definition of double sequence $a_{mn}$ being convergent to $l$? I have this definition. Definition: The double sequence $(a_{m,n})^∞_{m,n=1}$ is said to Converge to the real number $A∈ \mathbb R$ if for all $ϵ>0$ there exists an $N∈ \mathbb N$ such that if $m,n≥N$ then $∣a_{m,n}−A∣<...

If $(s(n,m))$ is a double sequence such that (i) the iterated limit lim$_{m \to \infty} $(lim$ _{n \to \infty }s(n,m)$) = a, and (ii) the limit (lim$ _{n \to \infty }s(n,m)$) exists uniformly for every $m \in \mathbb N$, then the double limit lim$ _{n,m \to \infty }s(n,m)=a$. Can anyone p...