# Proving that $\min\limits_{x \neq 0} \frac{\|Ax\|_2}{\|x\|_2}=\sigma_n$

For $A \in \mathbb R^{m \times n}, m \geq n, x \in \mathbb R^{n \times 1}$, how to prove $\min\limits_{x \neq 0} \frac{\|Ax\|_2}{\|x\|_2}=\sigma_n$? As I'm new to SVD, can anyone help me in solving this problem?

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Do you know the corresponding proof for $\max$ and $\sigma_1$? – J. M. Sep 17 '11 at 13:07
Im sorry. i dont know that proof. Proving any one of them is better – Learner Sep 17 '11 at 13:10
What have you tried and where are you stuck? If you substitute $A = U \Sigma V^T$ where the right-hand side is the SVD of $A$, then what is an equivalent form of the problem? – cardinal Sep 17 '11 at 13:22