We will prove this by induction on $n$ for all proper morphisms. \medskip\noindent If $n = 0$, then $f$ is a finite morphism (More on Morphisms, Lemma \ref{more-morphisms-lemma-characterize-finite}) and the result is true by Proposition \ref{proposition-finite-higher-direct-image-zero}. \medskip\noindent If $n > 0$, then using Lemma \ref{lemma-proper-base-change-stalk} we see that it suffices to prove $H^i_\etale(X, \mathcal{F}) = 0$ for $i > 2n$ and $X$ a proper scheme, $\dim(X) \leq n$ over an algebraically closed field $k$ and $\mathcal{F}$ is a torsion abelian sheaf on $X$. \medskip\noindent If $n = 1$ this follows from Theorem \ref{theorem-vanishing-curves}. Assume $n > 1$. By Proposition \ref{proposition-topological-invariance} we may replace $X$ by its reduction. Let $\nu : X^\nu \to X$ be the normalization. This is a surjective birational finite morphism (see Varieties, Lemma \ref{varieties-lemma-normalization-locally-algebraic}) and hence an isomorphism over a dense open $U \subset X$ (Morphisms, Lemma \ref{morphisms-lemma-birational-birational}). Then we see that $c : \mathcal{F} \to \nu_*\nu^{-1}\mathcal{F}$ is injective (as $\nu$ is surjective) and an isomorphism over $U$. Denote $i : Z \to X$ the inclusion of the complement of $U$. Since $U$ is dense in $X$ we have $\dim(Z) < \dim(X) = n$. By Proposition \ref{proposition-closed-immersion-pushforward} have $\Coker(c) = i_*\mathcal{G}$ for some abelian torsion sheaf $\mathcal{G}$ on $Z_\etale$. Then $H^q_\etale(X, \Coker(c)) = H^q_\etale(Z, \mathcal{F})$ (by Proposition \ref{proposition-finite-higher-direct-image-zero} and the Leray spectral sequence) and by induction hypothesis we conclude that the cokernel of $c$ has cohomology in degrees $\leq 2(n - 1)$. Thus it suffices to prove the result for $\nu_*\nu^{-1}\mathcal{F}$. As $\nu$ is finite this reduces us to showing that $H^i_\etale(X^\nu, \nu^{-1}\mathcal{F})$ is zero for $i > 2n$. This case is treated in the next paragraph. \medskip\noindent Assume $X$ is integral normal proper scheme over $k$ of dimension $n$. Choose a nonconstant rational function $f$ on $X$. The graph $X' \subset X \times \mathbf{P}^1_k$ of $f$ sits into a diagram $$ X \xleftarrow{b} X' \xrightarrow{f} \mathbf{P}^1_k $$ Observe that $b$ is an isomorphism over an open subscheme $U \subset X$ whose complement is a closed subscheme $Z \subset X$ of codimension $\geq 2$. Namely, $U$ is the domain of definition of $f$ which contains all codimension $1$ points of $X$, see Morphisms, Lemmas \ref{morphisms-lemma-rational-map-from-reduced-to-separated} and \ref{morphisms-lemma-extend-across} (combined with Serre's criterion for normality, see Properties, Lemma \ref{properties-lemma-criterion-normal}). Moreover the fibres of $b$ have dimension $\leq 1$ (as closed subschemes of $\mathbf{P}^1$). Hence $R^ib_*b^{-1}\mathcal{F}$ is nonzero only if $i \in \{0, 1, 2\}$ by induction. Choose a distinguished triangle $$ \mathcal{F} \to Rb_*b^{-1}\mathcal{F} \to Q \to \mathcal{F}[1] $$ Using that $\mathcal{F} \to b_*b^{-1}\mathcal{F}$ is injective as before and using what we just said, we see that $Q$ has nonzero cohomology sheaves only in degrees $0, 1, 2$ sitting on $Z$. Moreover, these cohomology sheaves are torsion by Lemma \ref{lemma-torsion-direct-image}. By induction we see that $H^i(X, Q)$ is zero for $i > 2 + 2\dim(Z) \leq 2 + 2(n - 2) = 2n - 2$. Thus it suffices to prove that $H^i(X', b^{-1}\mathcal{F}) = 0$ for $i > 2n$. At this point we use the morphism $$ f : X' \to \mathbf{P}^1_k $$ whose fibres have dimension $< n$. Hence by induction we see that $R^if_*b^{-1}\mathcal{F} = 0$ for $i > 2(n - 1)$. We conclude by the Leray spectral seqence $$ H^i(\mathbf{P}^1_k, R^jf_*b^{-1}\mathcal{F}) \Rightarrow H^{i + j}(X', b^{-1}\mathcal{F}) $$ and the fact that $\dim(\mathbf{P}^1_k) = 1$.