Inequality of max and min sum of eighen values of two symmetric matrices.

Question

Given two symmetric matrices A,B both are in $\mathbb{R}^{nxn}$. How can I show that $\lambda_{max}(A+B)\leq\lambda_{max}(A)+\lambda_{max}(B)$
and
$\lambda_{min}(A+B)\geq\lambda_{min}(A)+\lambda_{min}(B)$
using Rayleigh-Ritz Inequality.

Answer 1

Hint: $$ \max_{\|x\| = 1} x^T(A + B)x = \max_{\|x\| = 1} (x^TAx + x^TBx) \leq \max_{\|x\| = 1} x^TAx + \max_{\|x\| = 1} x^TBx $$