$ || \lambda(A) - \lambda(B) ||_p \prec_k || \lambda(A -B) ||_p$?

Question

Given two Hermitian matrices $\mathbf{A}$ and $\mathbf{B}$ and eigenvalue function $\lambda(\cdot)$ which returns eigenvalues of a matrix in non-increasing order.

I found the following is true from the page 44 of the paper "Some inequalities for sum and product of positive semidenite matrices. Linear Algebra and its Applications 293 (1999) 39-49":

$\lambda(A) - \lambda(B) \prec \lambda(A-B)$ and $ | \lambda(A) - \lambda(B) | \prec_k | \lambda(A -B) |$,

where $\prec_k$ stands for weak majorization.

My questions are:

1. Could anyone point out how to prove the above inequalities?
2. Is there some result on $ || \lambda(A) - \lambda(B) ||_p \prec_k || \lambda(A -B) ||_p$ where $|| \cdot ||_p$ denotes the $\ell_p$ norm.

Thanks for the help!