Since $X$ is quasi-compact we see that the special fibre $X_k$ is quasi-compact. Since the singularities of $X_k$ are at-worst-nodal, we see that $X_k$ has a finite number of nodes and is otherwise smooth over $k$. As $X \to \Spec(R)$ is flat with smooth generic fibre it follows that $X$ is smooth over $R$ except at the finite number of nodes of $X_k$ (use Morphisms, Lemma \ref{morphisms-lemma-smooth-at-point}). It follows that $X$ is regular at every point except for possibly the nodes of its special fibre (see Algebra, Lemma \ref{algebra-lemma-regular-goes-up}). Let $x \in X$ be such a node. Choose a diagram $$ \xymatrix{ X \ar[d] & U \ar[r] \ar[d] \ar[l] & \Spec(A) \ar[dl] \\ \Spec(R) & \Spec(R') \ar[l] } $$ as in Lemma \ref{lemma-etale-local-at-worst-nodal}. Observe that the case $A = R'[u, v]/(uv)$ cannot occur, as this would mean that the generic fibre of $X/R$ is singular (tiny detail omitted). Thus $A = R'[u, v]/(uv - \pi^n)$ for some $n \geq 0$. Since $x$ is a singular point, we have $n \geq 2$, see discussion in Example \ref{example-blowup}. \medskip\noindent After shrinking $U$ we may assume there is a unique point $u \in U$ mapping to $x$. Let $w \in \Spec(A)$ be the image of $u$. We may also assume that $u$ is the unique point of $U$ mapping to $w$. Since the two horizontal arrows are \'etale we see that $u$, viewed as a closed subscheme of $U$, is the scheme theoretic inverse image of $x \in X$ and the scheme theoretic inverse image of $w \in \Spec(A)$. Since blowing up commutes with flat base change (Divisors, Lemma \ref{divisors-lemma-flat-base-change-blowing-up}) we find a commutative diagram $$ \xymatrix{ X' \ar[d] & U' \ar[l] \ar[d] \ar[r] & W' \ar[d] \\ X & U \ar[l] \ar[r] & \Spec(A) } $$ with cartesian squares where the vertical arrows are the blowing up of $x, u, w$ in $X, U, \Spec(A)$. The scheme $W'$ was described in Example \ref{example-blowup}. We saw there that $W'$ at-worst-nodal of relative dimension $1$ over $R'$. Thus $W'$ is at-worst-nodal of relative dimension $1$ over $R$ (Algebraic Curves, Lemma \ref{curves-lemma-nodal-family-postcompose-etale}). Hence $U'$ is at-worst-nodal of relative dimension $1$ over $R$ (see Algebraic Curves, Lemma \ref{curves-lemma-nodal-family-etale-local-source}). Since $X' \to X$ is an isomorphism over the complement of $x$, we conclude the same thing is true of $X'/R$ (by Algebraic Curves, Lemma \ref{curves-lemma-nodal-family-etale-local-source} again). \medskip\noindent Finally, we need to argue that after doing a finite number of these blowups we arrive at a regular model $X_m$. This is rather clear because the ``invariant'' $n$ decreases by $2$ under the blowup described above, see computation in Example \ref{example-blowup}. However, as we want to avoid precisely defining this invariant and establishing its properties, we in stead argue as follows. If $n = 2$, then $W'$ is regular and hence $X'$ is regular at all points lying over $x$ and we have decreased the number of singular points of $X$ by $1$. If $n > 2$, then the unique singular point $w'$ of $W'$ lying over $w$ has $\kappa(w) = \kappa(w')$. Hence $U'$ has a unique singular point $u'$ lying over $u$ with $\kappa(u) = \kappa(u')$. Clearly, this implies that $X'$ has a unique singular point $x'$ lying over $x$, namely the image of $u'$. Thus we can argue exactly as above that we get a commutative diagram $$ \xymatrix{ X'' \ar[d] & U'' \ar[l] \ar[d] \ar[r] & W'' \ar[d] \\ X' & U' \ar[l] \ar[r] & W' } $$ with cartesian squares where the vertical arrows are the blowing up of $x', u', w'$ in $X', U', W'$. Continuing like this we get a compatible sequence of blowups which stops after $\lfloor n/2 \rfloor$ steps. At the completion of this process the scheme $X^{(\lfloor n/2 \rfloor)}$ will have one fewer singular point than $X$. Induction on the number of singular points completes the proof.