Entries versus Frobenius norm

Idea. A sum of $n^2$ non-negative numbers is at least its largest term and at most $n^2$ times its largest term.

Step 1: left inequality. Fix $(k,l)$. Since all terms are non-negative,
$$a_{kl}^2\le\sum_{i,j}a_{ij}^2=\|A\|_F^2.$$
Taking square roots, $|a_{kl}|\le\|A\|_F$, and then take the maximum over $(k,l)$.

Step 2: right inequality. Let $m=\max_{i,j}|a_{ij}|$. Then each $a_{ij}^2\le m^2$ and there are $n^2$ terms, so
$$\|A\|_F^2\le n^2m^2,\qquad\|A\|_F\le nm.$$

Equality cases. The left inequality is an equality iff $A$ has at most one non-zero entry (e.g. $A=E_{11}$). The right one is an equality iff $|a_{ij}|=m$ for all $i,j$ (e.g. the all-ones matrix, for which $\|J\|_F=n$ and $m=1$).

Remark. These inequalities show directly that the Frobenius norm and the max-entry norm are equivalent. They are the simplest instance of the fact (proved later) that all norms on $M_n(\mathbb{R})$ are equivalent. In particular a sequence of matrices converges iff each entry converges.