[proofplan]
We use the definition of the wave front set as the complement of the set of microlocally smooth pairs. A pair $(x_0,\xi_0)$ is microlocally smooth if some cutoff equal to $1$ near $x_0$ has a rapidly decaying localized [Fourier transform](/page/Fourier%20Transform) in a conic neighbourhood of $\xi_0$. Conicity follows because conic neighbourhoods are invariant under positive rescaling of the covector. Closedness follows by proving that microlocal smoothness is an open condition in $U \times \mathbb{R}^n_0$.
[/proofplan]
[step:Recall the microlocal smoothness criterion defining the complement of the wave front set]
Let $\mathcal{M}(u) \subset U \times \mathbb{R}^n_0$ denote the set of microlocally smooth pairs of $u$. Thus $(x_0,\xi_0) \in \mathcal{M}(u)$ means that there exist a cutoff function $\chi \in C_c^\infty(U)$, an open neighbourhood $W \subset U$ of $x_0$, and an open conic set $\Gamma \subset \mathbb{R}^n_0$ containing $\xi_0$ such that $\chi = 1$ on $W$ and the Fourier transform of $\chi u$ decays rapidly on $\Gamma$.
More explicitly, regarding $\chi u$ as a compactly supported distribution on $\mathbb{R}^n$ by extension by zero, define its Fourier transform as the smooth map
\begin{align*}
\mathcal{F}(\chi u): \mathbb{R}^n &\to \mathbb{C}
\end{align*}
given by the usual Fourier transform of a compactly supported distribution. The rapid decay condition is that for every integer $N \ge 1$ there exists a constant $C_N > 0$ such that
\begin{align*}
|\mathcal{F}(\chi u)(\xi)| \le C_N(1 + |\xi|)^{-N}
\end{align*}
for every $\xi \in \Gamma$.
By definition,
\begin{align*}
\operatorname{WF}(u) = (U \times \mathbb{R}^n_0) \setminus \mathcal{M}(u).
\end{align*}
Therefore it is enough to prove that $\mathcal{M}(u)$ is open and conic.
[/step]
[step:Show that microlocal smoothness is invariant under positive rescaling of covectors]
Let $(x_0,\xi_0) \in \mathcal{M}(u)$, and let $\lambda > 0$. Choose $\chi \in C_c^\infty(U)$, an open neighbourhood $W \subset U$ of $x_0$, and an open conic set $\Gamma \subset \mathbb{R}^n_0$ containing $\xi_0$ satisfying the rapid decay condition above. Since $\Gamma$ is conic and $\lambda > 0$, we have $\lambda \xi_0 \in \Gamma$. The same cutoff $\chi$, the same base neighbourhood $W$, and the same conic neighbourhood $\Gamma$ therefore prove that $(x_0,\lambda \xi_0) \in \mathcal{M}(u)$.
Thus $\mathcal{M}(u)$ is conic in the covector variable. Applying the same statement with the positive scalar $\lambda^{-1}$ shows that if $(x_0,\lambda \xi_0) \in \mathcal{M}(u)$, then $(x_0,\xi_0) \in \mathcal{M}(u)$. Hence the complement $\operatorname{WF}(u)$ is also conic: for every $(x_0,\xi_0) \in \operatorname{WF}(u)$ and every $\lambda > 0$, one has $(x_0,\lambda \xi_0) \in \operatorname{WF}(u)$.
[guided]
Fix a microlocally smooth pair $(x_0,\xi_0)$. By definition, this means that there is a cutoff $\chi \in C_c^\infty(U)$, an open neighbourhood $W \subset U$ of $x_0$, and an open conic set $\Gamma \subset \mathbb{R}^n_0$ containing $\xi_0$ such that $\chi = 1$ on $W$ and, for every integer $N \ge 1$, there is a constant $C_N > 0$ with
\begin{align*}
|\mathcal{F}(\chi u)(\xi)| \le C_N(1 + |\xi|)^{-N}
\end{align*}
for all $\xi \in \Gamma$.
Now let $\lambda > 0$. The only issue is whether the same cone also works for $\lambda \xi_0$. Since $\Gamma$ is conic, membership is invariant under multiplication by positive scalars. Therefore $\xi_0 \in \Gamma$ implies $\lambda \xi_0 \in \Gamma$. Nothing else changes: the cutoff is still equal to $1$ near $x_0$, and the rapid decay estimates are still valid on the same set $\Gamma$. Hence $(x_0,\lambda \xi_0)$ is microlocally smooth.
This proves that the microlocally smooth set $\mathcal{M}(u)$ is conic in the covector variable. To transfer conicity to the wave front set, use the complement relation
\begin{align*}
\operatorname{WF}(u) = (U \times \mathbb{R}^n_0) \setminus \mathcal{M}(u).
\end{align*}
If $(x_0,\xi_0) \in \operatorname{WF}(u)$ and $\lambda > 0$, but $(x_0,\lambda \xi_0) \notin \operatorname{WF}(u)$, then $(x_0,\lambda \xi_0) \in \mathcal{M}(u)$. Applying the already proved conicity of $\mathcal{M}(u)$ with the positive scalar $\lambda^{-1}$ gives $(x_0,\xi_0) \in \mathcal{M}(u)$, contradicting $(x_0,\xi_0) \in \operatorname{WF}(u)$. Therefore $(x_0,\lambda \xi_0) \in \operatorname{WF}(u)$.
[/guided]
[/step]
[step:Prove that microlocal smoothness is open in $U \times \mathbb{R}^n_0$]
Let $(x_0,\xi_0) \in \mathcal{M}(u)$. Choose $\chi \in C_c^\infty(U)$, an open neighbourhood $W \subset U$ of $x_0$, and an open conic set $\Gamma \subset \mathbb{R}^n_0$ containing $\xi_0$ such that $\chi = 1$ on $W$ and
\begin{align*}
|\mathcal{F}(\chi u)(\eta)| \le C_N(1 + |\eta|)^{-N}
\end{align*}
for every $\eta \in \Gamma$ and every integer $N \ge 1$, with a constant $C_N > 0$ depending on $N$.
Because $W$ is open in $U$, the product set
\begin{align*}
W \times \Gamma \subset U \times \mathbb{R}^n_0
\end{align*}
is an open neighbourhood of $(x_0,\xi_0)$ in the relative topology of $U \times \mathbb{R}^n_0$.
Let $(x,\xi) \in W \times \Gamma$. Since $W$ is an open neighbourhood of $x$ and $\chi = 1$ on $W$, the cutoff $\chi$ is equal to $1$ near $x$. Since $\Gamma$ is an open conic neighbourhood of $\xi$, and the same rapid decay estimates hold for $\mathcal{F}(\chi u)$ on all of $\Gamma$, the same data $\chi$, $W$, and $\Gamma$ prove that $(x,\xi) \in \mathcal{M}(u)$. Hence
\begin{align*}
W \times \Gamma \subset \mathcal{M}(u).
\end{align*}
Thus every point of $\mathcal{M}(u)$ has an open neighbourhood contained in $\mathcal{M}(u)$, so $\mathcal{M}(u)$ is open in $U \times \mathbb{R}^n_0$.
[/step]
[step:Conclude that the wave front set is closed and conic]
We have proved that $\mathcal{M}(u)$ is open in $U \times \mathbb{R}^n_0$. Since
\begin{align*}
\operatorname{WF}(u) = (U \times \mathbb{R}^n_0) \setminus \mathcal{M}(u),
\end{align*}
the set $\operatorname{WF}(u)$ is closed relative to $U \times \mathbb{R}^n_0$. We also proved that $\operatorname{WF}(u)$ is invariant under positive rescaling in the covector variable. Therefore $\operatorname{WF}(u)$ is a closed conic subset of $U \times \mathbb{R}^n_0$.
[/step]
