$$\mathcal S(\Bbb R):=\big\{f:\Bbb R\mapsto\Bbb R\ :\ \forall k,l\in\Bbb N \ \sup\limits_{x\in\Bbb R}\{|x^k|\cdot|f^{(l)}(x)|\}<\infty\big\}$$
I am trying to show that $\mathcal S(\Bbb R)\subset L^p(\Bbb R)$ for any $p\in[1,\infty]$
First if $p=\infty$ then $\|f\|_{\infty}=\sup\limits_{t\in\Bbb R}|f(t)|<\infty$ by taking $k=0,\ l=0$ in the definition.
If $p\in[1,\infty)$
I would like to prove something pretty strong: If $f\in\mathcal S(\Bbb R)$ then for any $\epsilon>0\ \exists q\in\Bbb R$ s.t. $$\int\limits_{-\infty}^{-q}|f(x)|dx+\int\limits_{q}^{+\infty}|f(x)|dx<\epsilon$$ that will be stronger than what's necessary but I think it is true for functions in $\mathcal S(\Bbb R)$. In fact those function go to zero really fast as $|x|\rightarrow\infty$, faster than any polynomial (I'm pretty sure).
Hint: choose for a fixed $p$ an integer $k$ such that the function $t\mapsto \left(1+\left\lvert t\right\vert^k\right)^{-1}$ belongs to $\mathbb L^p$ and observe that $$\left\lvert f(t)\right\rvert^{p}\leqslant\left(1+\left\lvert t\right\vert^k\right)^{-p}\left( \sup_{x\in\mathbb R}\left(1+\left\lvert x\right\vert^k\right)\left\lvert f(x)\right\rvert\right)^{p}.$$