I'm reading through these notes, and in particular, the proof of Lemma $4.1.9$ on page 3 of the pdf.
The space $C_c^{\infty}(\mathbb{R}^n)$ is dense in $S(\mathbb{R}^n)$.
They define, 
and later estimate 
I don't understand: How do they obtain the two inequalities for their estimate?
Looking back a few weeks later, I think these are the steps:
\begin{align} |(\phi_j(x)-1)\partial^{\alpha}f(x)| &=|(\phi_j(x)-1)| \cdot | \partial^{\alpha} f(x)| \\ & \leq (1+\underbrace{\sup |\phi|}_{=1}) \cdot | \partial^{\alpha} f(x)| \\ & \leq (1+\underbrace{\sup |\phi|}_{=1}) \cdot (1+j)^{-1}(1+||x||)|\partial^{\alpha}f(x)| \\ &\phantom{} \text{ the above obtained by noting that $(1+||x||)\cdot(1+j)^{-1} \geq 1$} \text{ because $||x|| \geq j$ }\\ &\leq 2 \cdot \underbrace{(1+j)^{-1}}_{\leq j^{-1}}(1+||x||)|\partial^{\alpha}f(x)|\\ & \leq 2 j^{-1} (1+||x||)|\partial^{\alpha}f(x)|. \end{align}