The following is supposed to be a proof that the Schwartz space is contained in $L^p(\mathbb{R}^n)$ for $1 \leq p <+\infty$:
Given $f \in \mathcal{S}(\mathbb{R}^n),$ there is a constant $K>0$ such that $$|f(x)| \leq \frac{K}{1+|x|^2}$$ for every $x \in \mathbb{R}^n.$ Hence, $$|f(x)|^p \leq \frac{K^p}{(1+|x|^2)^p} \leq \frac{K^p}{1+|x|^2} \in L^1(\mathbb{R}^n)$$ and we conclude that $f \in L^p(\mathbb{R}^n),$ as desired.
Now, I don't know what $|x|^2$ means here. If it is the norm, then the bound for $|f(x)|$ is true, but for $n=2$ the function $1/(1+|x|^2) = 1/(1+x_1^2+x_2^2)$ is not integrable on $\mathbb{R}^2.$ Help will be appreciated.