Suppose that $A$ is an $m\times n$ matrix and let $\text{min}(m,\,n)$ denote the minimum of $m$ and $n$. Prove that $\rank{A}\leq \text{min}(m,\,n)$. (If $m$ and $n$ are two numbers, then $\text{min}(m,\,n)$ stands for the number that is the smaller of the two. For example $\text{min}(4,\,6)=4$.)