A fixed-coordinate point-blowup chain defines a formal arc

The directed union. By [F1], each $B_{n+1}$ is a localization of a chart $B_n[{\mathfrak{m}}_n\mathrm{/}f]$ for a nonzero $f\in {\mathfrak{m}}_n$. Such a chart is a subring of $\mathrm{Frac}(B_n)$, so its localization is a domain containing $B_n$ and with the same fraction field. Thus all $B_n$ embed compatibly in $\mathrm{Frac}(B)$, and locality gives ${\mathfrak{m}}_{n+1}\cap B_n={\mathfrak{m}}_n$. The union $U={\bigcup }_n{B}_n$ is a local domain with maximal ideal $\mathfrak{n}={\bigcup }_n{\mathfrak{m}}_n$: this union is a proper ideal, and every element outside it is already a unit in a ring $B_n$. The compatible residue-field identifications give $U\mathrm{/}\mathfrak{n}=\kappa$. Moreover $\mathfrak{m}\mathrm{\ne }0$, since blowing up the zero ideal has no points by [F1]; injectivity and $\mathfrak{m}{B}_1=t{B}_1$ therefore imply $t\mathrm{\ne }0$.

The maximal ideal is generated by $t$. For $k>n$ the hypothesis gives ${\mathfrak{m}}_n{B}_k=t{B}_k$, hence ${\mathfrak{m}}_{n}U=tU$. Since extending each ${\mathfrak{m}}_n$ to $U$ stays inside $\mathfrak{n}$, their union is still $\mathfrak{n}$, and consequently $\mathfrak{n}=tU$. The element $t$ is a nonzero nonunit in the domain $U$.

Successive quotients. For $j\ge 0$, cancellation in $U$ identifies $U\mathrm{/}tU$ with $t^{j}U\mathrm{/}{t}^{j+1}U$ by multiplication by $t^j$. In particular $t^j\mathrm{\notin }{t}^{j+1}U$, since otherwise cancellation would make $t$ a unit. Thus $U\mathrm{/}{t}^{N}U$ has a filtration of length $N$ whose factors are $\kappa$; it is a local Artinian ring for every $N\ge 1$.

The inverse limit. Put , with projections ${\pi }_N:V\to U\mathrm{/}{t}^{N}U$. Each ${\pi }_N$ is surjective: a representative $u\in U$ of any prescribed class gives the compatible system of its residues. This alone does not identify the kernel; that identification follows from regularity of $t$ in $U$.

Compatible division and kernels. Fix $N\ge 1$. Multiplication by $t^N$ identifies $U\mathrm{/}{t}^{j}U$ with $t^{N}U\mathrm{/}{t}^{N+j}U$ for every $j\ge 1$, by cancellation in $U$. If $v=(v_j{)}_j\in \ker {\pi }_N$, then $v_{N+j}$ lies in $t^{N}U\mathrm{/}{t}^{N+j}U$; its unique quotient by $t^N$ defines $w_j\in U\mathrm{/}{t}^{j}U$. These quotients are compatible, so they define $w\in V$ with $t^{N}w=v$. Conversely $t^{N}V\subseteq \ker {\pi }_N$, proving $\ker {\pi }_N=t^{N}V$. Multiplication by $t^N$ is also injective on $V$: if $t^{N}w=0$, the component at $N+j$ says $w_{N+j}\in t^{j}U\mathrm{/}{t}^{N+j}U$, whose reduction gives $w_j=0$ for every $j$. Thus division in $V$ is unique whenever it is defined, without having assumed that $V$ is a domain.

Completeness and locality. Steps 4.1 and 5.1 give $V\mathrm{/}{t}^{N}V\cong U\mathrm{/}{t}^{N}U$ for every $N$, so is $t$-adically complete and ${\bigcap }_N{t}^{N}V={\bigcap }_N\ker {\pi }_N=0$. Also $V\mathrm{/}tV=\kappa$, making $tV$ a proper ideal. If $v\mathrm{\notin }tV$, choose a unit $u\in U$ lifting its nonzero residue; its image in $V$ is a unit, and $v=u(1+\epsilon )$ with $\epsilon \in tV$. The series ${\sum }_{j\ge 0}(-\epsilon {)}^j$ converges in $V$ and inverts $1+\epsilon$. Thus every element outside $tV$ is a unit, and $V$ is local with maximal ideal $tV$ and residue field $\kappa$.

Orders and the domain property. For $0\mathrm{\ne }v\in V$, separatedness gives a largest $N\ge 0$ with $v\in t^{N}V$. Write $v=t^{N}u$; maximality makes $u\mathrm{\notin }tV$, hence a unit. For two such elements, regularity of powers of $t$ proved in step 5.1 shows that their product is nonzero and has order the sum of their orders: any further divisibility by $t$ would, after cancellation, put a product of units in $tV$. Therefore $V$ is a domain.

A complete DVR. Every nonzero ideal $J\subseteq V$ has an element of least order $N$. Writing it as $t^{N}u$ with $u$ a unit shows that $t^N\in J$, and all other elements have order at least $N$, so $J=(t^N)$. Hence every ideal is principal and $V$ is Noetherian. Its only prime ideals are $0$ and $tV$: powers $(t^N)$ with $N>1$ are not prime. Thus $V$ has dimension one, and its maximal ideal is generated by $t$, so it is regular and is a DVR by [F4]. Completeness and its residue field were established in step 6.1.

The arc and its centres. The maps $B_n\to U\to V$ are local and induce the given residue-field identifications. Since ${\mathfrak{m}}_{n}U=tU$, one has ${\mathfrak{m}}_{n}V=tV$. The map $B\to V$ is therefore a formal arc, nonsingular because the image of $t\in \mathfrak{m}$ generates $tV\mathrm{/}{t}^{2}V$. For every $n$, the pullback of the closed-point ideal of $\mathrm{Spec}{B}_n$ is the effective Cartier divisor cut out by $t$. The universal property [F2] gives a unique factorization through its blowup. The given local map $B_{n+1}\to V$, viewed through its chart in [F1], gives such a factorization; locality places its closed point at the prescribed centre. Uniqueness therefore identifies the successive centres of the arc with the given chain.

Completion and surjectivity. Since $\mathfrak{m}V=tV$, there are compatible maps $B\mathrm{/}{\mathfrak{m}}^N\to V\mathrm{/}{t}^{N}V$. Passing to inverse limits extends $B\to V$ uniquely to $\widehat{B}\to V$, using [F3] and step 6.1. Given $v\in V$, set $r_0=v$ and successively choose $c_i\in B$ with residue equal to that of $r_i$, then define $r_{i+1}=(r_i-c_i)\mathrm{/}t$. Existence and uniqueness of this division follow from step 5.1; [F5] supplies the sequence of residue lifts. The partial sums $s_N={\sum }_{i<N}{c}_i{t}^i$ satisfy $v-s_N=t^N{r}_N$ in $V$. They are also Cauchy in $B$ for its $\mathfrak{m}$-adic topology, since $s_M-s_N\in {\mathfrak{m}}^N$ whenever $M\ge N$, as $t\in \mathfrak{m}$. Hence they define an element of $\widehat{B}$ whose image agrees with $v$ modulo every $t^{N}V$, and separatedness gives equality. Thus $\widehat{B}\to V$ is surjective, without a coefficient-field choice.