The theorem is:
Let $(a_{m,n})_{m,n}$ be a double sequence that converges to a limit L. Assume that the sequence $(a_{m,n})_{n}$ converges. Then $\lim_{m\to\infty}(\lim_{n\to\infty}a_{m,n})=L$.
Would the converse be true? Where if $\lim_{m\to\infty}(\lim_{n\to\infty}a_{m,n})=L$, then the sequence $(a_{m,n})_{n}$ converges?
No, consider $a_{m,n}=\dfrac{m}{m+n}$, then $\lim_{m\rightarrow\infty}(\lim_{n\rightarrow\infty}a_{m,n})=\lim_{m\rightarrow\infty}(0)=0$.
Assume that $a_{m,n}\rightarrow L$ for some $L$, then $a_{m,m}\rightarrow L$ and $a_{m,2m}\rightarrow L$. But $a_{m,m}=\dfrac{1}{2}\rightarrow\dfrac{1}{2}$ and $a_{m,2m}=\dfrac{1}{3}\rightarrow\dfrac{1}{3}$, a contradiction.