Is there an infinite matrix $A_{mn}$ such that $\lim\limits_{n \to \infty }A_{mn}=0 $ for every $m$ and $\lim\limits_{m \to \infty }A_{mn}=1 $ for every $n$ ?
Any clue as to how to start on this?
Is there an infinite matrix $A_{mn}$ such that $\lim\limits_{n \to \infty }A_{mn}=0 $ for every $m$ and $\lim\limits_{m \to \infty }A_{mn}=1 $ for every $n$ ?
Any clue as to how to start on this?
How about $A_{mn}=\left(\frac{m}{m+1}\right)^n$? We get
$\lim_{n\to\infty}A_{mn}=\lim_{n\to\infty}\left(\frac{m}{m+1}\right)^n=0$ because $0<\frac{m}{m+1}<1$ for all $m\geq 1$, and $\lim_{m\to\infty}A_{mn}=\lim_{m\to\infty}\left(\frac{m}{m+1}\right)^n=\left(\lim_{m\to\infty}\frac{m}{m+1}\right)^n=1^n=1.$
Another example: