Inequality regarding norm of a positive definite matrix

Since $\rm A$ is symmetric and positive definite, it has a symmetric and positive definite square root $\mathrm A^\frac 12$ and $\lambda_{\max} \left( \mathrm A \right) = \| \mathrm A\|_2$, i.e., the spectral radius is equal to the spectral norm. Let $\rm v := \mathrm A^\frac 12 u$. Hence,

$$\dfrac{\mathrm u^\top \mathrm A^2 \mathrm u}{\mathrm u^\top \mathrm A \,\, \mathrm u} = \dfrac{\mathrm u^\top \mathrm A^\frac 12 \mathrm A \, \mathrm A^\frac 12 \mathrm u}{\mathrm u^\top \mathrm A^\frac 12 \mathrm A^\frac 12 \mathrm u} = \dfrac{\mathrm v^\top \mathrm A \, \mathrm v}{\mathrm v^\top \mathrm v} \leq \lambda_{\max} \left( \mathrm A \right) = \| \mathrm A\|_2$$