Suppose group $G$ continuously acts on $X$ top.space. I wanted to define some kind of completion of $G$ with respect to $X$, so for each sequence $g_i$, if $g_{i} x$ converges for every $x\in X$, I defined $$ \epsilon:X→X \\ x \mapsto \lim_{n\to\infty} g_{n} x. $$ I want to claim that the collection of such maps gives the natural completion $\bar G$ of $G$ with repect to $X$, that is, $G$ is open dense in $\bar G$ (with the topology that makes $g\mapsto gx$ continuous for each $x\in X$) and the right action of $G$ on itself naturally extends to $\bar G$. (I am not yet sure if these are true or not.)
I came up with this idea because I wanted to naturally extend $\mathbb{R^*}^n$ action on $\mathbb{R}^n$, where $(a_1,\ldots,a_n)(x_1,\ldots,x_n)=(a_1 x_1,\ldots, a_n x_n)$ to $\mathbb{R}^n$ action, which is not a group with multiplicative operation.
So my first step was to prove that $$ \epsilon:X→X \\ x \mapsto \lim_{n\to\infty} g_{n} x. $$ is continuous, but I kept failing to prove this statement. Is this statement generally not true? Do I need any assumption, such as G being ableian or X being Hausdorff, to make it true?