In this exercice $C_{c}^\infty$ denotes the set of all functions defined in $\mathbb{R}^n$ that are infinitely differentiable and compactly supported.
For this proof it is suggested to use the following statements:
1. If $\|x\|$ denotes the Euclidean norm of $x\in\mathbb{R}^n$ and if $a>0$, then exists $c>0$ depending only on $n$ and $a$ such that $$c^{-1}\left(|x_1|^\alpha+\cdots +|x_n|^\alpha\right)\leq \|x\|^\alpha\leq c\left(|x_1|^\alpha+\cdots +|x_n|^\alpha\right)$$.
2. $\displaystyle{\int_{B(0,1)}\frac{1}{\|x\|^\alpha}\,dx<\infty\Longleftrightarrow}\hspace{.1cm} a<n\qquad$ and $\qquad\displaystyle{\int_{B(0,1)^c}\frac{1}{\|x\|^\alpha}\,dx<\infty\Longleftrightarrow}\hspace{.1cm} a>n.$
(Here $B(0,1)$ denote the unit ball in $\mathbb{R}^n$).
Let's see found that $\mathcal{S}\subset L^1\cap L^\infty$. Let $f\in\mathcal{S}$. Since $f$ is bounded on $\mathbb{R}^n$, so will it be $\quad x_{j}^{n+1}f\quad$ for $j=1,...,n$. Also, because $ f $ is bounded, there is a constant $ c $ such that $$\left(1+|x_1|^{n+1}+\cdots +|x_n|^{n+1}\right)|f(x)|\leq c.$$ by statement 1 there is another constant $c'$ such that $$|f(x)|\leq \frac{c'}{1+\|x\|^{n+1}}$$ and for the and by statement 2, $f\in L^1$.
My questions are about how to test the claims used in this demo. Also I have not been able to demonstrate the inclusion $C_{c}^\infty\subset \mathcal{S}$.