Pringsheim convergence

Remarks

Example: $a_{m,n} = \frac{1}{m} + \frac{1}{n}$ converges to $0$, because $m, n \ge N$ gives $\abs{a_{m,n}} \le \frac{2}{N}$, which is less than $\varepsilon$ once $N > 2/\varepsilon$. Non-example: $a_{m,n} = \frac{m}{m+n}$ has no double limit, because $a_{N,N} = \frac12$ while $a_{N,n} \to 0$ as $n \to \infty$, so no single $L$ works for $\varepsilon = \frac14$. A Pringsheim convergent double sequence need not be bounded: put $a_{1,n} = n$ and $a_{m,n} = 0$ for $m \ge 2$; the double limit is $0$, yet the first row diverges. So the double limit may exist while an iterated limit $\lim_m \lim_n a_{m,n}$ does not; the Moore–Osgood theorem gives conditions under which they agree.