I am dealing with a problem on the existence of the limit for a sequence of function $\{g_n(x),n\in\mathbb N\}$ with function $g_n(x):\mathbb R^n\to\mathbb R^m$, i.e. to judge the existence of limit $g_\infty(x)$ on a compact set $\mathcal X\subset\mathbb R^n$.
I consider Cauchy criterion because I known the upper bound of the difference between $g_n(x)$ and $g_{n-1}(x)$, for $x\in\mathcal X$.
Let $f_n(x)=g_n(x)-g_{n-1}(x)\le h_n(x)=A^nx$ with $g_0(x)=0$, for $x\in \mathcal X$. Here, the inequality of vector is to be performed elementwise.
$A$ is a $n\times n$ matrix whose eigenvalues $\rho(A)$ are contained in the interior of the unit disc, i.e. $\rho(A)<1$ and $\lim_{i\to \infty}A^i=0$.
Therefore, $h_n(x)=A^nx$ may converge pointwise and uniformly to $0$ for all $x\in\mathcal X$.
My question. Does $g_\infty(x)=\sum_{n=1}^{\infty}f_n(x)$ converge pointwise/uniformly on $\mathcal X$?