If every elements of a matrix convergence in probability, do this matrices converge in probability?

Question

Let $\widehat{\sum}=\left(a_{i,j}^{N}\right)_{1\leq i\leq j\leq N}$ and $\sum=\left(a_{i,j}\right)_{1\leq i\leq j\leq N}$.

If $a_{i,j}^{N}\overset{P}{\to} a_{i,j}$ for any $1\leq i\leq j\leq N$, then $\widehat{\sum}\overset{P}{\to} \sum$? Is this true or are there additional conditions required?

Answer 1

This is true! No further conditions required here. The given condition is enough to prove that the maximum difference of the entries with the limiting matrix converge in probability to zero, and every norm on matrices is equivalent (in terms of convergence) with the sup norm.