Let $v_1, ..., v_k$ be linearly independent vectors in $\mathbb{R}^n$ and $k < n$. Let $\sigma$ be the parallelpiped formed by these $k$ vectors inside $\mathbb{R}^n$. Then as described in Understanding the volume of $k$-parallelpiped in $\mathbb{R}^n$ $$ vol(\sigma)=\sqrt{\mid \det A^T A \mid} $$ with matrix $A = [v_1 ... v_k]$.
I've been trying to prove that $$ vol(\sigma) \leq \prod_{i=1}^k || v_i||_2. $$ My strategy was to use induction, but the determinant expression is quite complicated and it had not been helpful. Any suggestion on how I can prove this is appreciated!