I am proving or disproving the following statement: Let $f_n$ be a sequence of Schwartz functions in $\mathbb R^d$, such that $f_n$ converges to 0 uniformly. Is it true that $f_n$ converges to 0 in $L^p(\mathbb R^d)$ for all $1 \leq p\leq \infty$?
My idea: The case $p=\infty$ is automatically true, and for simplicity we consider the case $p=1$ only. Let $\epsilon>0$, and by uniform convergence, there is an $N$ such that for all $n\geq N$, for all $x\in \mathbb R^d$, $$ |f_n(x)|<\epsilon $$ Notice that $f_n$ is Schwartz, and hence integrable. There is an $M_n>0$ such that we have: $$ \int_{|x|>M_n}|f_n(x)|<\epsilon $$ But the problem lies in whether $\sup_n M_n<\infty$. If this is true, then we are done.
If on the contrary this is false, please give a counterexample. Thank you!
Edited: If the above is hard to prove, please try to prove my original question, assuming $f_n$ converges to 0 in the metric of Schwartz space.
Given $f\in\mathcal{S}(\mathbb{R}^d)$ and $a>0$ let $f_n(x)=n^{-a}\,f(x/n)$. Then $f_n$ converges uniformly to $0$ but $\|f_n\|_p=n^{d/p-a}\,\|f\|_p$ does not converge to $0$ if $1\le p\le d/a$.
If $f_n$ converges to $0$ in the metric of $\mathcal{S}(\mathbb{R}^d)$, then $$ \lim_{n\to\infty}\sup_{x\in\mathbb{R}^d}(1+|x|)^{d+1}\,|f_n(x)|=0. $$ From this it is easy to see that $\lim_{n\to\infty}\|f_n\|_p=0$ for all $p\ge1$.