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 $∣a_{m,n}−A∣<ϵ$ and we say $A$ is the Double Limit of this double sequence written lim$_{m,n→∞}a_{m,n}=A$. If no such $A∈ \mathbb R$ satisfies this, then we say that the the double sequence $(a_{m,n})^∞_{m,n=1}$ diverges. I took help from here.
If I go by this definition then convergent double sequence $(a_{m,n})^∞_{m,n=1}$ may not be bounded. Example :
$a_{1n} = n$, $ a_{mn} = 1/m + 1/n$ for all $m \geq 2 $ and $n\in \mathbb N$
It seems odd to me. I feel that I am going wrong anywhere. Can anyone please tell me where I am being wrong?
Suppose the sequence $(m,n) \mapsto a_{mn}$ converges in the Pringsheim sense. This means there exists $L$ such that for any $\epsilon > 0$ there exists $N$ such that $|a_{mn} - L| < \epsilon$ for all $m,n \geqslant N$.
With convergence in this sense, it is not necessary that the set $\{a_{mn}: m,n \in \mathbb{N}\}$ be bounded. Your example illustrates this.
On the other hand a convergent double sequence is bounded eventually. That is, there exists $N \in \mathbb{N}$ such that the set $\{a_{mn}: m \geqslant N,n \geqslant N\}$is bounded.