Three-epsilon (triangle-inequality chaining) argument

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Problem 4.1.3 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 4.1 "Topology of $\reals[n]$"

Let {x(i,j)∣i,j∈N}\left\{ x(i,j) \mid i, j \in \mathbb {N}\right\} be a doubly indexed set in a complete metric space (X,ρ)(X, \rho ) such that

ρ(x(i,j),x(k,ℓ))≤min⁡{max⁡{1i,1k},max⁡{1j,1ℓ}}. \rho (x(i,j), x(k,\ell )) \leq \min \left\{ \max \left\{ \frac{1}{i}, \frac{1}{k}\right\} , \max \left\{ \frac{1}{j}, \frac{1}{\ell }\right\} \right\} .

Prove that the iterated limits lim⁡i→∞lim⁡j→∞x(i,j)\lim_{i \to \infty } \lim_{j \to \infty } x(i,j), lim⁡j→∞lim⁡i→∞x(i,j)\lim_{j \to \infty } \lim_{i \to \infty } x(i,j) exist and are equal.

?

References

[deSi04] de Souza P.N., Silva J. Berkeley Problems in Mathematics. 3rd ed., Springer, 2004.