Double sequence, if $(x_m)_m$ and $(y_n)_n$ converge, then they have the same limit?

Question

Let $(a_{m, n})_{m, n}$ be a "double sequence" of real numbers; that is, for every pair $(m, n) \in \mathbb{N} \times \mathbb{N}$, $a_{m, n}$ is a real number. Suppose that, for every $m$, $(a_{m, n})_n$ converges to a limit $x_m$, and that for every $n$ the sequence $(a_{m, n})_m$ converges to a limit $y_n$. Suppose that for every $\epsilon > 0$ there exists $N \in \mathbb{N}$ such that $|a_{m, n} - x_m| < \epsilon$ and $|a_{m, n} - y_n| < \epsilon$ for every $m$, $n > N$. Does it follow that if $(x_m)_m$ and $(y_n)_n$ converge, then they have the same limit?

Answer 1

Since you can make the difference between $x_m$ and $y_n$ arbitrarily small for large $m$ and $n$, if one of the sequences converges to a limit $a$ then so does the other:

$$\forall k>N:|x_k-y_k|\leq|x_k-a_{k,k}|+|a_{k,k}-y_k|<2\epsilon.$$