I am reading Krantz's A Primer of Real Analytic Functions. Let $\sum_{\mu}a_{\mu}x^{\mu}$ be a power series in several variables (where we're using the multi-index notation) with the domain of convergence $$\mathcal{C} = \{x\in\mathbb{R}^{m}: \text{there exists }r > 0\text{ such that }\sum_{\mu}|a_{\mu}y^{\mu}| < \infty\text{ for all }y\in B_{r}(x)\}.$$ We need to show that the series converges uniformly on compact subsets of $\mathcal{C}$.
My attempt: For $(x_{1}, \dots, x_{m})\in\mathbb{R}^{m}$ the silhouette $s(x)$ is defined by $$s(x) = \{(r_{1}x_{1}, \dots, r_{m}x_{m}): r_{i}\in(-1, 1)\text{ for all }i = 1, \dots, m\}.$$ It is shown earlier in the book that if $x$ is a point sucht that $\{|a_{\mu}x^{\mu}|\}_{\mu}$ is bounded and $D$ is a compact subset of $s(x)$ then the series converges uniformly and absolutely on $D$.
Let $K$ be a compact subset of $\mathcal{C}$. Let $x\in K$ and pick $r > 0$ such that $\sum_{\mu}|a_{\mu}y^{\mu}| < \infty$ for all $y\in B_{r}(x)$. Pick a point $(1+\delta)x+z\in B_{r}(x)$ (by suitably choosing $\delta > 0$ and $z\in\mathbb{R}^{m}$) such that $z_{i}\neq 0$ only when $x_{i} = 0$. We claim that there exists $\varepsilon > 0$ such that the closed cube $\{y\in\mathbb{R}^{m} : \max_{i}|y_{i}-x_{i}|\leq \varepsilon\}$ is contained in $s(x)$. To find such an $\varepsilon$, it suffices to ensure that $$x_{i}+\varepsilon < (1+\delta)|x_{i}|+|z_{i}|\:\:\:\:\text{and}\:\:\:\:x_{i}-\varepsilon > -(1+\delta)|x_{i}|-|z_{i}|$$ for all $i = 1, \dots, m$. One can choose $$0 < \varepsilon < \min\{(|x_{i}|\pm x_{i})+\delta|x_{i}|+|z_{i}| : i = 1, \dots, m\}.$$ Now we may cover $K$ by finitely many such closed cubes and use the above convergence result.
Could anyone tell me if this approach seems correct and if there are better ways to go about this proof? Any help is appreciated.
