Show that $ \ \lambda \leq \|A^TA\|$.

Question

Let $ \lambda $ be the eigenvalue of $A^TA$, $A \in \mathbb{R}^{n \times n}$.

Let $\|\cdot\|$ be a norm on $\mathbb{R}^n$ with associated subordinate matrix $\|\cdot\|$ on $\mathbb{R}^{n \times n}$.

Show that $ \ \lambda \leq \|A^TA\|$.

If $x \neq 0$, then $(A^TA) x=\lambda x \Rightarrow ||A^TA||=\lambda$.

But I have to show $||A^TA|| \geq \lambda$.