eigenvalues of two positive commutative matrices

eigenvalues of two positive commutative matrices

Let $A$ and $B$ be two positive commutative matrices. I am going to prove
$$\lambda_{j}(A+B)\leq \lambda_{j}(A)+\lambda_{j}(B)$$
for $j=1,2,\ldots n$, where $\lambda_{j}$ are eigenvalues of matrix and
these eigenvalues are in a decreasing order as follows
$$\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{n}$$ I heard that
$\lambda_{max}$ or $\lambda_{1}$ is correct for the first inequality, I
mean $\lambda_{1}(A+B)\leq \lambda_{1}(A)+\lambda_{1}(B)$ ?
Can we take out the max eigenvalue and say the similar thing for others?
Is it true for operators which may have infinite eigenvalues?
Thanks in advanced