Show that convergence of a sequence of power-series is equivalent to convergence of their respective coefficients.
More precisely, let $\mathbf{a}:=\left[a_{i,j}\right]_{i,j\in\mathbb{N_0}}$ be a real matrix, such as a stochastic matrix, such that
i) $\forall i\in\mathbb{N}_0,\space\mathbf{L}\leq\mathbf{a}_i$ for some real row vector $\mathbf{L}$ that sums to a real number ($\mathbf{a}_i$ being $\mathbf{a}$'s $i$th row),
ii) $\sup_{i\in\mathbb{N}_0} \sum \mathbf{a}_i<\infty$ (Given a real vector $\mathbf{v}=\left[v_m\right]_{m\in\mathbb{N}_0}$, we define $\sum\mathbf{v}:=\sum_{m=0}^\infty v_m$ whenever the series on the right converges, possibly to $\pm\infty$.)
Additionally, let $\mathbf{b}:=\left[b_j\right]_{j\in\mathbb{N}_0}$ be a real row vector that sums to a finite number and define $\mathbf{x}:=\left[x_j\right]_{j\in\mathbb{N}_0}$ to be the column vector of monomial functions: $$x_j:\left(-1,1\right)\rightarrow\left(-1,1\right),\space\space x_j\left(y\right):=y^j$$
Lastly, set $\mathbf{f}:=\mathbf{a}\mathbf{x}$, $F:=\mathbf{b}\mathbf{x}$ and $$\mathbf{S}:=\left\{S\subseteq\left(-1,1\right):\space S\mathrm{\, has\, an\, accumulation\, point\, }\in \left(-1,1\right)\right\}$$ So $\mathbf{f}$ is a column vector whose components are power series and $F$ is a power series, and the radii of convergence of all these power series are $\geq1$.
Show that $$\begin{align}\mathbf{a}_i\underset{i\rightarrow\infty}{\rightarrow}\mathbf{b} &\iff f_i\underset{i\rightarrow\infty}{\rightarrow}F\mathrm{\, on\, }\left(-1,1\right)\\ &\iff\exists S\in\mathbf{S},\space f_i\underset{i\rightarrow\infty}{\rightarrow}F\mathrm{\, on\, }S\end{align}$$
Notes
We take $0^0$ to be $1$, so $x_0\equiv 1$ on $\left(-1, 1\right)$.
Conditions $\left(i\right)$ and $\left(ii\right)$ above will be required for the application of Fatou's lemma in the proof. In $\left(ii\right)$, $\sup$ is used rather than the laxer $\limsup$ required by Fatou's lemma, to ensure that the $f_i$'s radii of convergence are $\geq1$.
The claim is adapted from Klenke, where an equivalent claim is stated without proof (Lemma 3.6, $\left(i\right)\iff\left(iii\right)\iff\left(iv\right)$).