Show there exists a sequence $\{f_n\}$ in the Schwartz space $S(\mathbb R)$ with limit $f$ for which $ \lim \|f_n\|_{u,v} \text{ induced that } f \not\in S(\mathbb R) \text{ for some } u,v. $ But $ \lim \|f_n\|_{a,b} \text{ does not converges when } a \ne u,b\ne v. $
Use the standard norm on $S(\mathbb R)$ $ \|f\|_{a,b}= \sup_{x \in \mathbb R} |x^af^{(b)}(x)| ,\, a,b \in \mathbb Z_+. $
This post shows $\sqrt{f(x)}$ does not belongs to $S(\mathbb R)$ when $f(x)=e^{-x^2} \left(e^{-x^2}+\sin ^2(x)\right)$.
I guess one way is to make $\|f_n\|_{u,v}$ unbounded, while keeping the other derivative bounded.
But I don't see how to do that.