If double indexed sequence $\{a_{m,n}\}$ has the same limit along each diagonal, i.e., $$ \lim_{m,n\to\infty\\ m/n\to c} a_{m,n} = a $$ for each $c\in(0,\infty)$, it is not necessarily true that the double limit $\lim_{m,n\to\infty}a_{m,n}$ exists. For example, consider $a_{m,n} = m/n^2$.
However, I wonder if things might get different if $c\in[0,\infty]$. That is, if $$ \lim_{m,n\to\infty\\ m/n\to c} a_{m,n} = a $$ for $c\in[0,\infty]$. Do we now have $\lim_{m,n\to\infty}a_{m,n} = a$? It seems correct but I cannot prove it. Any hint or a counterexample?