Throughout this proof we will use that $i_*$ is an exact functor, and hence that $Ri_* = i_*$, see Modules, Lemma \ref{modules-lemma-i-star-exact}. \medskip\noindent Assume $E$ is $m$-pseudo-coherent. Let $x \in X$. We will find a neighbourhood of $x$ such that $i_*E$ is $m$-pseudo-coherent on it. If $x \not \in Z$ then this is clear. Thus we may assume $x \in Z$. We will use that $U \cap Z$ for $x \in U \subset X$ open form a fundamental system of neighbourhoods of $x$ in $Z$. After shrinking $X$ we may assume $E$ is bounded above. We will argue by induction on the largest integer $p$ such that $H^p(E)$ is nonzero. If $p < m$, then there is nothing to prove. If $p \geq m$, then $H^p(E)$ is an $\mathcal{O}_Z$-module of finite type, see Cohomology, Lemma \ref{cohomology-lemma-finite-cohomology}. Thus we may choose, after shrinking $X$, a map $\mathcal{O}_Z^{\oplus n}[-p] \to E$ which induces a surjection $\mathcal{O}_Z^{\oplus n} \to H^p(E)$. Choose a distinguished triangle $$ \mathcal{O}_Z^{\oplus n}[-p] \to E \to C \to \mathcal{O}_Z^{\oplus n}[-p + 1] $$ We see that $H^j(C) = 0$ for $j \geq p$ and that $C$ is $m$-pseudo-coherent by Cohomology, Lemma \ref{cohomology-lemma-cone-pseudo-coherent}. By induction we see that $i_*C$ is $m$-pseudo-coherent on $X$. Since $i_*\mathcal{O}_Z$ is $m$-pseudo-coherent on $X$ as well, we conclude from the distinguished triangle $$ i_*\mathcal{O}_Z^{\oplus n}[-p] \to i_*E \to i_*C \to i_*\mathcal{O}_Z^{\oplus n}[-p + 1] $$ and  Cohomology, Lemma \ref{cohomology-lemma-cone-pseudo-coherent} that $i_*E$ is $m$-pseudo-coherent. \medskip\noindent Assume that $i_*E$ is $m$-pseudo-coherent. Let $z \in Z$. We will find a neighbourhood of $z$ such that $E$ is $m$-pseudo-coherent on it. We will use that $U \cap Z$ for $z \in U \subset X$ open form a fundamental system of neighbourhoods of $z$ in $Z$. After shrinking $X$ we may assume $i_*E$ and hence $E$ is bounded above. We will argue by induction on the largest integer $p$ such that $H^p(E)$ is nonzero. If $p < m$, then there is nothing to prove. If $p \geq m$, then $H^p(i_*E) = i_*H^p(E)$ is an $\mathcal{O}_X$-module of finite type, see Cohomology, Lemma \ref{cohomology-lemma-finite-cohomology}. Choose a complex $\mathcal{E}^\bullet$ of $\mathcal{O}_Z$-modules representing $E$. We may choose, after shrinking $X$, a map $\alpha : \mathcal{O}_X^{\oplus n}[-p] \to i_*\mathcal{E}^\bullet$ which induces a surjection $\mathcal{O}_X^{\oplus n} \to i_*H^p(\mathcal{E}^\bullet)$. By adjunction we find a map $\alpha : \mathcal{O}_Z^{\oplus n}[-p] \to \mathcal{E}^\bullet$ which induces a surjection $\mathcal{O}_Z^{\oplus n} \to H^p(\mathcal{E}^\bullet)$. Choose a distinguished triangle $$ \mathcal{O}_Z^{\oplus n}[-p] \to E \to C \to \mathcal{O}_Z^{\oplus n}[-p + 1] $$ We see that $H^j(C) = 0$ for $j \geq p$. From the distinguished triangle $$ i_*\mathcal{O}_Z^{\oplus n}[-p] \to i_*E \to i_*C \to i_*\mathcal{O}_Z^{\oplus n}[-p + 1] $$ the fact that $i_*\mathcal{O}_Z$ is pseudo-coherent and  Cohomology, Lemma \ref{cohomology-lemma-cone-pseudo-coherent} we conclude that $i_*C$ is $m$-pseudo-coherent. By induction we conclude that $C$ is $m$-pseudo-coherent. By Cohomology, Lemma \ref{cohomology-lemma-cone-pseudo-coherent} again we conclude that $E$ is $m$-pseudo-coherent.