Assume that the double sequence $(a_{m,n})_{m,n}$ converges to $0$. Then is the set $\left \{ a_{m,n}:m,n\in\mathbb{N} \right \}$ bounded?

Question

Assume that the double sequence $(a_{m,n})_{m,n}$ converges to $0$. Then is the set $\left \{ a_{m,n}:m,n\in\mathbb{N} \right \}$ bounded?

Is the set is not bounded because the set is for all $m,n\in \mathbb{N}$, and not for all $m,n\geq N\in \mathbb{N}$? Does a counterexample work to prove this statement is wrong?

Answer 1

\begin{align*} (a_{1,1},a_{1,2},a_{1,3},a_{1,4},...)&=(1,1/2,1/3,1/4,...)\\ (a_{2,1},a_{2,2},a_{2,3},a_{2,4},...)&=(2,1/2,1/3,1/4,...)\\ (a_{3,1},a_{3,2},a_{3,3},a_{3,4},...)&=(3,1/2,1/3,1/4,...)\\ \vdots \end{align*} The sequence $\{a_{m,n}\}$ tends to $0$ as $m,n\rightarrow\infty$ but it is not bounded, as can be seen in the first column of the above arrays.