Suppose we are given the following inequality, which holds for any real number $s$ and $x,y \in \mathbb{R}^n$:
\begin{equation} (1 + |x + y|)^s \leq (1 + |x|)^s(1 + |y|)^{|s|} \end{equation}
(This is sometimes called Peetre's Inequality). How can I use this to show that, for a given integer $k$ and real number $d$, and $\xi,\eta \in \mathbb{R}^n$
\begin{equation} (1 + |\xi|)^{d}(1 + |\xi - \eta|)^{-k}(1 + |\eta|)^{-k} \leq (1 + |\xi|)^{|d|-k}(1 + |\eta|)^{|d| - k} \end{equation}
This is stated in some notes I am currently studying to learn about Fourier Analysis. I cannot fill in the details, I always end up with more terms than there ought to be .. any help would be great !!