Let $T \subset |X|$ be a closed subset with $X \setminus T \to X$ quasi-compact. Let $r_U$ be the integer of Definition \ref{definition-approximation} adapted to the pair $(U, T \cap |U|)$. Set $T' = T \setminus |U|$. Endow $T'$ with the induced reduced subspace structure. Since $|T'|$ is contained in $|X| \setminus |U|$ we see that $j^{-1}(T') \to T'$ is an isomorphism. Moreover, $V \setminus j^{-1}(T')$ is quasi-compact as it is the fibre product of $U \times_X V$ with $X \setminus T$ over $X$ and we've assumed $U \times_X V$ quasi-compact and $X \setminus T \to X$ quasi-compact. Let $r'$ be the number of affines needed to cover $V \setminus j^{-1}(T')$. We claim that $r = \max(r_U, r')$ works for the pair $(X, T)$. \medskip\noindent To see this choose a triple $(T, E, m)$ such that $E$ is $(m - r)$-pseudo-coherent and $H^i(E)$ is supported on $T$ for $i \geq m - r$. Let $t$ be the largest integer such that $H^t(E)|_U$ is nonzero. (Such an integer exists as $U$ is quasi-compact and $E|_U$ is $(m - r)$-pseudo-coherent.) We will prove that $E$ can be approximated by induction on $t$. \medskip\noindent Base case: $t \leq m - r'$. This means that $H^i(E)$ is supported on $T'$ for $i \geq m - r'$. Hence Lemma \ref{lemma-approximation-affine} guarantees the existence of an approximation $P \to E|_V$ of $(T', E|_V, m)$ on $V$. Applying Lemma \ref{lemma-open} we see that $(T', E, m)$ can be approximated. Such an approximation is also an approximation of $(T, E, m)$. \medskip\noindent Induction step. Choose an approximation $P \to E|_U$ of $(T \cap |U|, E|_U, m)$. This in particular gives a surjection $H^t(P) \to H^t(E|_U)$. In the rest of the proof we will use the equivalence of Lemma \ref{lemma-derived-quasi-coherent-small-etale-site} (and the compatibilities of Remark \ref{remark-match-total-direct-images}) for the representable algebraic spaces $V$ and $U \times_X V$. We will also use the fact that $(m - r)$-pseudo-coherence, resp.\ perfectness on the Zariski site and \'etale site agree, see Lemmas \ref{lemma-descend-pseudo-coherent} and \ref{lemma-descend-perfect}. Thus we can use the results of Derived Categories of Schemes, Section \ref{perfect-section-lift} for the open immersion $U \times_X V \subset V$. In this way Derived Categories of Schemes, Lemma \ref{perfect-lemma-lift-perfect-complex-plus-shift-support} implies there exists a perfect object $Q$ in $D(\mathcal{O}_V)$ supported on $j^{-1}(T)$ and an isomorphism $Q|_{U \times_X V} \to (P \oplus P[1])|_{U \times_X V}$. By Derived Categories of Schemes, Lemma \ref{perfect-lemma-lift-map} we can replace $Q$ by $Q \otimes^\mathbf{L} I$ and assume that the map $$ Q|_{U \times_X V} \longrightarrow (P \oplus P[1])|_{U \times_X V} \longrightarrow P|_{U \times_X V} \longrightarrow E|_{U \times_X V} $$ lifts to $Q \to E|_V$. By Lemma \ref{lemma-glue} we find an morphism $a : R \to E$ of $D(\mathcal{O}_X)$ such that $a|_U$ is isomorphic to $P \oplus P[1] \to E|_U$ and $a|_V$ isomorphic to $Q \to E|_V$. Thus $R$ is perfect and supported on $T$ and the map $H^t(R) \to H^t(E)$ is surjective on restriction to $U$. Choose a distinguished triangle $$ R \to E \to E' \to R[1] $$ Then $E'$ is $(m - r)$-pseudo-coherent (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cone-pseudo-coherent}), $H^i(E')|_U = 0$ for $i \geq t$, and $H^i(E')$ is supported on $T$ for $i \geq m - r$. By induction we find an approximation $R' \to E'$ of $(T, E', m)$. Fit the composition $R' \to E' \to R[1]$ into a distinguished triangle $R \to R'' \to R' \to R[1]$ and extend the morphisms $R' \to E'$ and $R[1] \to R[1]$ into a morphism of distinguished triangles $$ \xymatrix{ R \ar[r] \ar[d] & R'' \ar[d] \ar[r] & R' \ar[d] \ar[r] & R[1] \ar[d] \\ R \ar[r] &  E \ar[r] & E' \ar[r] & R[1] } $$ using TR3. Then $R''$ is a perfect complex (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-two-out-of-three-perfect}) supported on $T$. An easy diagram chase shows that $R'' \to E$ is the desired approximation.