We are going to use Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cech-vanish-collection} to prove this. Let $\mathcal{B} \subset \Ob(X_\proetale)$ be the set of affine objects $U$ of $X_\proetale$ such that $\mathcal{O}(X) \to \mathcal{O}(U)$ is ind-\'etale. Let $\text{Cov}$ be the set of pro-\'etale coverings $\{U_i \to U\}_{i = 1, \ldots, n}$ with $U \in \mathcal{B}$ such that $\mathcal{O}(U) \to \mathcal{O}(U_i)$ is ind-\'etale for $i = 1, \ldots, n$. Properties (1) and (2) of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cech-vanish-collection} hold for $\mathcal{B}$ and $\text{Cov}$ by Lemmas \ref{lemma-composition-ind-etale}, \ref{lemma-base-change-ind-etale}, and \ref{lemma-proetale-affine} and Proposition \ref{proposition-weakly-etale}. \medskip\noindent To check condition (3) suppose that $\mathcal{U} = \{U_i \to U\}_{i = 1, \ldots, n}$ is an element of $\text{Cov}$. We have to show that the higher Cech cohomology groups of $\epsilon^{-1}\mathcal{I}$ with respect to $\mathcal{U}$ are zero. First we write $U_i = \lim_{a \in A_i} U_{i, a}$ as a directed inverse limit with $U_{i, a} \to U$ \'etale and $U_{i, a}$ affine. We think of $A_1 \times \ldots \times A_n$ as a direct set with ordering $(a_1, \ldots, a_n) \geq (a_1', \ldots, a_n')$ if and only if $a_i \geq a_i'$ for $i = 1, \ldots, n$. Observe that $\mathcal{U}_{(a_1, \ldots, a_n)} = \{U_{i, a_i} \to U\}_{i = 1, \ldots, n}$ is an \'etale covering for all $a_1, \ldots, a_n \in A_1 \times \ldots \times A_n$. Observe that $$ U_{i_0} \times_U U_{i_1} \times_U \ldots \times_U U_{i_p} = \lim_{(a_1, \ldots, a_n) \in A_1 \times \ldots \times A_n} U_{i_0, a_{i_0}} \times_U U_{i_1, a_{i_1}} \times_U \ldots \times_U U_{i_p, a_{i_p}} $$ for all $i_0, \ldots, i_p \in \{1, \ldots, n\}$ because limits commute with fibred products. Hence by Lemma \ref{lemma-limit-pullback} and exactness of filtered colimits we have $$ \check{H}^p(\mathcal{U}, \epsilon^{-1}\mathcal{I}) = \colim \check{H}^p(\mathcal{U}_{(a_1, \ldots, a_n)}, \epsilon^{-1}\mathcal{I})  $$ Thus it suffices to prove the vanishing for \'etale coverings of $U$! \medskip\noindent Let $\mathcal{U} = \{U_i \to U\}_{i = 1, \ldots, n}$ be an \'etale covering with $U_i$ affine. Write $U = \lim_{b \in B} U_b$ as a directed inverse limit with $U_b$ affine and $U_b \to X$ \'etale. By Limits, Lemmas \ref{limits-lemma-descend-finite-presentation}, \ref{limits-lemma-limit-affine}, and \ref{limits-lemma-descend-etale} we can choose a $b_0 \in B$ such that for $i = 1, \ldots, n$ there is an \'etale morphism $U_{i, b_0} \to U_{b_0}$ of affines such that $U_i = U \times_{U_{b_0}} U_{i, b_0}$. Set $U_{i, b} = U_b \times_{U_{b_0}} U_{i, b_0}$ for $b \geq b_0$. For $b$ large enough the family $\mathcal{U}_b = \{U_{i, b} \to U_b\}_{i = 1, \ldots, n}$ is an \'etale covering, see Limits, Lemma \ref{limits-lemma-descend-surjective}. Exactly as before we find that $$ \check{H}^p(\mathcal{U}, \epsilon^{-1}\mathcal{I}) = \colim \check{H}^p(\mathcal{U}_b, \epsilon^{-1}\mathcal{I}) = \colim \check{H}^p(\mathcal{U}_b, \mathcal{I}) $$ the final equality by Lemma \ref{lemma-fully-faithful}. Since each of the {\v C}ech complexes on the right hand side is acyclic in positive degrees (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-injective-trivial-cech}) it follows that the one on the left is too. This proves condition (3) of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cech-vanish-collection}. Since $X \in \mathcal{B}$ the lemma follows.