Properness of the blow ups of arithmetic schemes In Silverman's textbook Advanced Topics in the Arithmetic of Elliptic Curves, on page 345, he defined the blow up of an arithmetic surface (or a two dimensional scheme) at the point $(\pi,x,y) = (0,0,0)$ (i.e. reducing modulo $\pi$ since we are interested in what happens on the special fiber).
I believe we can extend his definition directly to blowing up subschemes e.g., say if we want to blow-up the whole $x$-axis instead, then we only consider the generators $\pi$ and $y$ for the subscheme? Also, his definition should not be limited to only arithmetic surfaces (we just add more variables and charts)?
Now let us fix an arithmetic scheme $C$ over a local ring $\mathcal{O}$ with uniformizer $\pi$. Following Silverman's definition, I suppose blowing up $C$ at any arithmetic subschemes (which we include the additional generator $\pi$ so that the generic fiber remains unaffected) gives us a new model $C^{\prime}$, together with a blow up map $C^{\prime} \rightarrow C$. My main question being, if $C$ is proper then is $C^{\prime}$ also proper? i.e. is the morphism $C^{\prime} \rightarrow \text{Spec}(\mathcal{O})$ a proper morphism?
 A: Yes, all of what you ask is true. Let $X$ be a scheme, and let $\mathcal{I}\subset\mathcal{O}_X$ be a quasi-coherent sheaf of ideals with corresponding closed subscheme $Z$. Then the blowup of $X$ along $Z$ is $$Bl_Z X := \underline{\operatorname{Proj}}_X \bigoplus_{n\geq 0} \mathcal{I}^n,$$ equipped with the natural projection $Bl_ZX\to X$. Being a relative proj, this morphism is automatically proper. In particular, writing $C'\to \operatorname{Spec}\mathcal{O}$ as $C'\to C\to\operatorname{Spec} \mathcal{O}$ and observing the composition of proper morphisms is proper, we see that $C'\to \operatorname{Spec}\mathcal{O}$ is indeed proper.
For more material about this definition of the blowup along a subscheme, you may consult Hartshorne chapter II section 7, Vakil chapter 22, or tag 01OF at the Stacks Project.

To see that this recovers the "classical" construction of the blowup, we work affine locally on $X$ and so we may take $X=\operatorname{Spec} A$. Then we may work with $I=\mathcal{I}(X)$ (the global sections of the sheaf of ideals) and the usual proj instead of the relative proj. Then $\bigoplus_{n\geq 0} I^n\cong A[tI]$ where $t$ is in degree 1 ($t$ is a formal variable to keep track of the grading). Choosing generators $f_1,\cdots,f_n$ for $I$, we can re-express $A[tI]$ as $A[y_1,\cdots,y_n]/(f_iy_j-f_jy_i)$ with the $f_i\in A$ and the $y_i$ in degree one (representing $tf_i$). But the Proj of this algebra is exactly the classical construction of the blowup: it's $V((f_iy_j-f_jy_i)_{i,j})\subset X\times\Bbb P^{n-1}$ where the $y_i$ are coordinates on $\Bbb P^{n-1}$.
