Let us first prove this in the quasi-compact case, since it is perhaps the most interesting case. In this case we produce inductively a sequence of blowups $$ X = X_0 \xleftarrow{b_0} X_1 \xleftarrow{b_1} X_2 \leftarrow \ldots $$ and finite sets of effective Cartier divisors $\{D_{n, i}\}_{i \in I_n}$. At each stage these will have the property that any triple intersection $D_{n, i} \cap D_{n, j} \cap D_{n, k}$ is empty. Moreover, for each $n \geq 0$ we will have $I_{n + 1} = I_n \amalg P(I_n)$ where $P(I_n)$ denotes the set of pairs of elements of $I_n$. Finally, we will have $$ b_n^{-1}(D_{n, i}) = D_{n + 1, i} + \sum\nolimits_{i' \in I_n, i' \not = i} D_{n + 1, \{i, i'\}} $$ We conclude that for each $n \geq 0$ we have $(b_0 \circ \ldots \circ b_n)^{-1}(D_i)$ is a nonnegative integer combination of the divisors $D_{n + 1, j}$, $j \in I_{n + 1}$. \medskip\noindent To start the induction we set $X_0 = X$ and $I_0 = I$ and $D_{0, i} = D_i$. \medskip\noindent Given $(X_n, \{D_{n, i}\}_{i \in I_n})$ let $X_{n + 1}$ be the blowup of $X_n$ in the closed subscheme $Z_n = \bigcup_{\{i, i'\} \in P(I_n)} D_{n, i} \cap D_{n, i'}$. Note that the closed subschemes $D_{n, i} \cap D_{n, i'}$ are pairwise disjoint by our assumption on triple intersections. In other words we may write $Z_n = \coprod_{\{i, i'\} \in P(I_n)} D_{n, i} \cap D_{n, i'}$. Moreover, in a Zariski neighbourhood of $D_{n, i} \cap D_{n, i'}$ the morphism $b_n$ is equal to the blowup of the scheme $X_n$ in the closed subscheme $D_{n, i} \cap D_{n, i'}$, and the results of Lemma \ref{lemma-two-divisors} apply. Hence setting $D_{n + 1, \{i, i'\}} = b_n^{-1}(D_i \cap D_{i'})$ we get an effective Cartier divisor. The Cartier divisors $D_{n + 1, \{i, i'\}}$ are pairwise disjoint. Clearly we have $b_n^{-1}(D_{n, i}) \supset D_{n + 1, \{i, i'\}}$ for every $i' \in I_n$, $i' \not = i$. Hence, applying Divisors, Lemma \ref{divisors-lemma-difference-effective-Cartier-divisors} we see that indeed $b^{-1}(D_{n, i}) = D_{n + 1, i} + \sum\nolimits_{i' \in I_n, i' \not = i} D_{n + 1, \{i, i'\}}$ for some effective Cartier divisor $D_{n + 1, i}$ on $X_{n + 1}$. In a neighbourhood of $D_{n + 1, \{i, i'\}}$ these divisors $D_{n + 1, i}$ play the role of the primed divisors of Lemma \ref{lemma-two-divisors}. In particular we conclude that $D_{n + 1, i} \cap D_{n + 1, i'} = \emptyset$ if $i \not = i'$, $i, i' \in I_n$ by part (6) of Lemma \ref{lemma-two-divisors}. This already implies that triple intersections of the divisors $D_{n + 1, i}$ are zero. \medskip\noindent OK, and at this point we can use the quasi-compactness of $X$ to conclude that the invariant \begin{equation} \label{equation-invariant} \epsilon(X, \{D_i\}_{i \in I}) = \max\{\epsilon_Z(D_i, D_{i'}) \mid Z \subset X, \dim_\delta(Z) = d - 1, \{i, i'\} \in P(I)\} \end{equation} is finite, since after all each $D_i$ has at most finitely many irreducible components. We claim that for some $n$ the invariant $\epsilon(X_n, \{D_{n, i}\}_{i \in I_n})$ is zero. Namely, if not then by Lemma \ref{lemma-two-divisors} we have a strictly decreasing sequence $$ \epsilon(X, \{D_i\}_{i \in I}) = \epsilon(X_0, \{D_{0, i}\}_{i \in I_0}) > \epsilon(X_1, \{D_{1, i}\}_{i \in I_1}) > \ldots $$ of positive integers which is a contradiction. Take $n$ with invariant $\epsilon(X_n, \{D_{n, i}\}_{i \in I_n})$ equal to zero. This means that there is no integral closed subscheme $Z \subset X_n$ and no pair of indices $i, i' \in I_n$ such that $\epsilon_Z(D_{n, i}, D_{n, i'}) > 0$. In other words, $\dim_\delta(D_{n, i}, D_{n, i'}) \leq d - 2$ for all pairs $\{i, i'\} \in P(I_n)$ as desired. \medskip\noindent Next, we come to the general case where we no longer assume that the scheme $X$ is quasi-compact. The problem with the idea from the first part of the proof is that we may get and infinite sequence of blowups with centers dominating a fixed point of $X$. In order to avoid this we cut out suitable closed subsets of codimension $\geq 3$ at each stage. Namely, we will construct by induction a sequence of morphisms having the following shape $$ \xymatrix{ X = X_0 \\ U_0 \ar[u]^{j_0} & X_1 \ar[l]_{b_0} \\  & U_1 \ar[u]^{j_1} & X_2 \ar[l]_{b_1} \\  & & U_2 \ar[u]^{j_2} & X_3 \ar[l]_{b_2} } $$ Each of the morphisms $j_n : U_n \to X_n$ will be an open immersion. Each of the morphisms $b_n : X_{n + 1} \to U_n$ will be a proper birational morphism of integral schemes. As in the quasi-compact case we will have effective Cartier divisors $\{D_{n, i}\}_{i \in I_n}$ on $X_n$. At each stage these will have the property that any triple intersection $D_{n, i} \cap D_{n, j} \cap D_{n, k}$ is empty. Moreover, for each $n \geq 0$ we will have $I_{n + 1} = I_n \amalg P(I_n)$ where $P(I_n)$ denotes the set of pairs of elements of $I_n$. Finally, we will arrange it so that $$ b_n^{-1}(D_{n, i}|_{U_n}) = D_{n + 1, i} + \sum\nolimits_{i' \in I_n, i' \not = i} D_{n + 1, \{i, i'\}} $$ \medskip\noindent We start the induction by setting $X_0 = X$, $I_0 = I$ and $D_{0, i} = D_i$. \medskip\noindent Given $(X_n, \{D_{n, i}\})$ we construct the open subscheme $U_n$ as follows. For each pair $\{i, i'\} \in P(I_n)$ consider the closed subscheme $D_{n, i} \cap D_{n, i'}$. This has ``good'' irreducible components which have $\delta$-dimension $d - 2$ and ``bad'' irreducible components which have $\delta$-dimension $d - 1$. Let us set $$ \text{Bad}(i, i') = \bigcup\nolimits_{W \subset D_{n, i} \cap D_{n, i'} \text{ irred.\ comp. with }\dim_\delta(W) = d - 1} W $$ and similarly $$ \text{Good}(i, i') = \bigcup\nolimits_{W \subset D_{n, i} \cap D_{n, i'} \text{ irred.\ comp. with }\dim_\delta(W) = d - 2} W. $$ Then $D_{n, i} \cap D_{n, i'} = \text{Bad}(i, i') \cup \text{Good}(i, i')$ and moreover we have $\dim_\delta(\text{Bad}(i, i') \cap \text{Good}(i, i')) \leq d - 3$. Here is our choice of $U_n$: $$ U_n = X_n \setminus \bigcup\nolimits_{\{i, i'\} \in P(I_n)} \text{Bad}(i, i') \cap \text{Good}(i, i'). $$ By our condition on triple intersections of the divisors $D_{n, i}$ we see that the union is actually a disjoint union. Moreover, we see that (as a scheme) $$ D_{n, i}|_{U_n} \cap D_{n, i'}|_{U_n} = Z_{n, i, i'} \amalg G_{n, i, i'} $$ where $Z_{n, i, i'}$ is $\delta$-equidimensional of dimension $d - 1$ and $G_{n, i, i'}$ is $\delta$-equidimensional of dimension $d - 2$. (So topologically $Z_{n, i, i'}$ is the union of the bad components but throw out intersections with good components.) Finally we set $$ Z_n = \bigcup\nolimits_{\{i, i'\} \in P(I_n)} Z_{n, i, i'} = \coprod\nolimits_{\{i, i'\} \in P(I_n)} Z_{n, i, i'}, $$ and we let $b_n : X_{n + 1} \to X_n$ be the blowup in $Z_n$. Note that Lemma \ref{lemma-two-divisors} applies to the morphism $b_n : X_{n + 1} \to X_n$ locally around each of the loci $D_{n, i}|_{U_n} \cap D_{n, i'}|_{U_n}$. Hence, exactly as in the first part of the proof we obtain effective Cartier divisors $D_{n + 1, \{i, i'\}}$ for $\{i, i'\} \in P(I_n)$ and effective Cartier divisors $D_{n + 1, i}$ for $i \in I_n$ such that $b_n^{-1}(D_{n, i}|_{U_n}) = D_{n + 1, i} + \sum\nolimits_{i' \in I_n, i' \not = i} D_{n + 1, \{i, i'\}}$. For each $n$ denote $\pi_n : X_n \to X$ the morphism obtained as the composition $j_0 \circ \ldots \circ j_{n - 1} \circ b_{n - 1}$. \medskip\noindent {\bf Claim:} given any quasi-compact open $V \subset X$ for all sufficiently large $n$ the maps $$ \pi_n^{-1}(V) \leftarrow \pi_{n + 1}^{-1}(V) \leftarrow \ldots $$ are all isomorphisms. Namely, if the map $\pi_n^{-1}(V) \leftarrow \pi_{n + 1}^{-1}(V)$ is not an isomorphism, then $Z_{n, i, i'} \cap \pi_n^{-1}(V) \not = \emptyset$ for some $\{i, i'\} \in P(I_n)$. Hence there exists an irreducible component $W \subset D_{n, i} \cap D_{n, i'}$ with $\dim_\delta(W) = d - 1$. In particular we see that $\epsilon_W(D_{n, i}, D_{n, i'}) > 0$. Applying Lemma \ref{lemma-two-divisors} repeatedly we see that $$ \epsilon_W(D_{n, i}, D_{n, i'}) < \epsilon(V, \{D_i|_V\}) - n $$ with $\epsilon(V, \{D_i|_V\})$ as in (\ref{equation-invariant}). Since $V$ is quasi-compact, we have $\epsilon(V, \{D_i|_V\}) < \infty$ and taking $n > \epsilon(V, \{D_i|_V\})$ we see the result. \medskip\noindent Note that by construction the difference $X_n \setminus U_n$ has $\dim_\delta(X_n \setminus U_n) \leq d - 3$. Let $T_n = \pi_n(X_n \setminus U_n)$ be its image in $X$. Traversing in the diagram of maps above using each $b_n$ is closed it follows that $T_0 \cup \ldots \cup T_n$ is a closed subset of $X$ for each $n$. Any $t \in T_n$ satisfies $\delta(t) \leq d - 3$ by construction. Hence $\overline{T_n} \subset X$ is a closed subset with $\dim_\delta(T_n) \leq d - 3$. By the claim above we see that for any quasi-compact open $V \subset X$ we have $T_n \cap V \not = \emptyset$ for at most finitely many $n$. Hence $\{\overline{T_n}\}_{n \geq 0}$ is a locally finite collection of closed subsets, and we may set $U = X \setminus \bigcup \overline{T_n}$. This will be $U$ as in the lemma. \medskip\noindent Note that $U_n \cap \pi_n^{-1}(U) = \pi_n^{-1}(U)$ by construction of $U$. Hence all the morphisms $$ b_n : \pi_{n + 1}^{-1}(U) \longrightarrow \pi_n^{-1}(U) $$ are proper. Moreover, by the claim they eventually become isomorphisms over each quasi-compact open of $X$. Hence we can define $$ U' = \lim_n \pi_n^{-1}(U). $$ The induced morphism $b : U' \to U$ is proper since this is local on $U$, and over each compact open the limit stabilizes. Similarly we set $J = \bigcup_{n \geq 0} I_n$ using the inclusions $I_n \to I_{n + 1}$ from the construction. For $j \in J$ choose an $n_0$ such that $j$ corresponds to $i \in I_{n_0}$ and define $D'_j = \lim_{n \geq n_0} D_{n, i}$. Again this makes sense as locally over $X$ the morphisms stabilize. The other claims of the lemma are verified as in the case of a quasi-compact $X$.