Let $a_{mn}$ be a sequence in some $\mathbb{R}^k$. We know in advance that

$$\lim_{n} ~a_{nn} = L_1, \qquad \lim_{m}~ \lim_{n} ~a_{mn} = L_2 $$

exist. Is there a sufficient criteria to conclude that $L_1 = L_2$? A simple example where this is not true is when $$a_{mn} = \frac{m}{n}.$$

Please note that I did not say $$ \lim_{n}~ \lim_{m} ~a_{mn} = L_3 $$ exists. If there is an answer available after assuming $L_3$ exists, then I would like to know that. But my original question doesn't assume $L_3$ exists.