Let $U \subset \mathbb{R}^n$ be open, and let $r: U \times \mathbb{R}^n \to \mathbb{C}$ be smooth. Suppose there exists a sequence $(m_N)_{N \in \mathbb{N}}$ in $\mathbb{R}$ such that $m_N \to -\infty$ as $N \to \infty$ and $r \in S^{m_N}_{1,0}(U \times \mathbb{R}^n)$ for every $N \in \mathbb{N}$. Then
In particular, if there exists $m_0 \in \mathbb{R}$ such that $r \in S^{m_0}_{1,0}(U \times \mathbb{R}^n)$ and $r \sim 0$ in the sense that $r \in S^{m_0-N}_{1,0}(U \times \mathbb{R}^n)$ for every $N \in \mathbb{N}$, then $r$ is smoothing.