Consider a function $f \in S(\mathbb{R})$ in the Schwartz space and define the sequence of functions $(f_n)_{n\in \mathbb{N}}$ by $f_n(x): = f(\frac{x}{n})$. I would like to show that $f_n$ does not converge uniformly toward $0$ as $n \to \infty$. (It was true with $x \mapsto \frac{x}{n} \cdot f(\frac{x}{n})$).
In order to do that, I supposed that the convergence is uniformly and I tried to find a contradiction with the convergence of the integrals, but without success.
Any suggestions? Thanks in advance.
You have $\|f_n\|_\infty = \|f\|_\infty$, so you have uniform convergence to zero if and only if $f \equiv 0$.