By Lemma \ref{lemma-regular-alteration-implies-local} the completion $A^\wedge$ of $A$ is normal. By Lemma \ref{lemma-port-regularity-to-completion} we see that $\Spec(A^\wedge)$ has a resolution. By Lemma \ref{lemma-normalized-blowup-completion} any sequence $Y_n \to Y_{n - 1} \to \ldots \to \Spec(A^\wedge)$ of normalized blowups of comes from a sequence of normalized blowups $X_n \to \ldots \to \Spec(A)$. Moreover if $Y_n$ is regular, then $X_n$ is regular by Lemma \ref{lemma-port-regularity-to-completion}. Thus it suffices to prove the lemma in case $A$ is complete. \medskip\noindent Assume in addition $A$ is a complete. We will use that $A$ is Nagata (Algebra, Proposition \ref{algebra-proposition-ubiquity-nagata}), excellent (More on Algebra, Proposition \ref{more-algebra-proposition-ubiquity-excellent}), and has a dualizing complex (Dualizing Complexes, Lemma \ref{dualizing-lemma-ubiquity-dualizing}). Moreover, the same is true for any ring essentially of finite type over $A$. If $B$ is a excellent local normal domain, then the completion $B^\wedge$ is normal (as $B \to B^\wedge$ is regular and More on Algebra, Lemma \ref{more-algebra-lemma-normal-goes-up} applies). We will use this without further mention in the rest of the proof. \medskip\noindent Let $X \to \Spec(A)$ be a resolution of singularities. Choose a sequence of normalized blowing ups $$ Y_n \to Y_{n - 1} \to \ldots \to Y_1 \to \Spec(A) $$ dominating $X$ (Lemma \ref{lemma-dominate-by-normalized-blowing-up}). The morphism $Y_n \to X$ is an isomorphism away from finitely many points of $X$. Hence we can apply Lemma \ref{lemma-dominate-by-blowing-up-in-points} to find a sequence of blowing ups $$ X_m \to X_{m - 1} \to \ldots \to X $$ in closed points such that $X_m$ dominates $Y_n$. Diagram $$ \xymatrix{ & Y_n \ar[rd] \ar[rr] & & \Spec(A) \\ X_m \ar[rr] \ar[ru] & & X \ar[ru] } $$ To prove the lemma it suffices to show that a finite number of normalized blowups of $Y_n$ produce a regular scheme. By our diagram above we see that $Y_n$ has a resolution (namely $X_m$). As $Y_n$ is a normal surface this implies that $Y_n$ has at most finitely many singularities $y_1, \ldots, y_t$ (because $X_m \to Y_n$ is an isomorphism away from the fibres of dimension $1$, see Varieties, Lemma \ref{varieties-lemma-modification-normal-iso-over-codimension-1}). \medskip\noindent Let $x_a \in X$ be the image of $y_a$. Then $\mathcal{O}_{X, x_a}$ is regular and hence defines a rational singularity (Lemma \ref{lemma-regular-rational}). Apply Lemma \ref{lemma-rational-propagates} to $\mathcal{O}_{X, x_a} \to \mathcal{O}_{Y_n, y_a}$ to see that $\mathcal{O}_{Y_n, y_a}$ defines a rational singularity. By Lemma \ref{lemma-rational-to-gorenstein} there exists a finite sequence of blowups in singular closed points $$ Y_{a, n_a} \to Y_{a, n_a - 1} \to \ldots \to \Spec(\mathcal{O}_{Y_n, y_a}) $$ such that $Y_{a, n_a}$ is Gorenstein, i.e., has an invertible dualizing module. By (the essentially trivial) Lemma \ref{lemma-equivalence-sequence-blowups} with $n' = \sum n_a$ these sequences correspond to a sequence of blowups $$ Y_{n + n'} \to Y_{n + n' - 1} \to \ldots \to Y_n $$ such that $Y_{n + n'}$ is normal and the local rings of $Y_{n + n'}$ are Gorenstein. Using the references given above we can dominate $Y_{n + n'}$ by a sequence of blowups $X_{m + m'} \to \ldots \to X_m$ dominating $Y_{n + n'}$ as in the following $$ \xymatrix{  & Y_{n + n'} \ar[rr] & & Y_n \ar[rd] \ar[rr] & & \Spec(A) \\ X_{m + m'} \ar[ru] \ar[rr] & & X_m \ar[rr] \ar[ru] & & X \ar[ru] } $$ Thus again $Y_{n + n'}$ has a finite number of singular points $y'_1, \ldots, y'_s$, but this time the singularities are rational double points, more precisely, the local rings $\mathcal{O}_{Y_{n + n'}, y'_b}$ are as in Lemma \ref{lemma-resolve-rational-double-points}. Arguing exactly as above we conclude that the lemma is true.