Let $A$ be an $m\times n$ real or complex matrix. Here it is shown that
$$\sigma_{min}(A)=\min_{x\in F^n, \|x\|=1}\|Ax\| \quad \quad (1)$$
where $\sigma_{min}(A)$ denotes the "least" singular value of $A$.
Usually this means $\sigma_{min}(A):=\sigma_{q}(A)$ with $q=\min\{m,n\}$. But then consider the case $n>m$:
Then $\text{rank(A)}\leq m<n$ so there exists $x\neq 0$ such that $Ax=0$, and WLOG we may assume $\|x\|=1$. Hence the right-hand side of $(1)$ is zero. But if $\text{rank(A)}= m$, then the left-hand side of $(1)$ is strictly positive.
Am I missing something? Do we need to define $\sigma_{min}(A):=\sigma_{n}(A)$ for the above to work?