Let $X$ be a Banach space and $H$ be a countably dense subset of $X$. Let $\{f_n\}$ be a sequence of function $f_n:X\to \mathbb{R}$ such that $$ f_n(x)\to f_x\qquad \forall x\in H $$ We suppose that $\{f_n\}$ is equicontinuous on $X$. Can we say that: for all $x\in X$ there exists $f_x\in \mathbb{R}$ such that
$$ f_n(x)\to f_x $$
Actually, the problem can be generalized as follows:
Let $X$ be a metric space and $Y$ a complete metric space. Let $H\subseteq X$ be a dense subset. For each $n\in\mathbb{N}$, let $f_{n}:X\rightarrow Y$ be a map. If:
(a) the family $\{f_{n}\mid n\in\mathbb{N}\}$ is equicontinuous, and
(b) for each $x\in H$, $\lim_{n}f_{n}(x)$ exists.
Then for each $x\in X$, $\lim_{n}f_{n}(x)$ exists.
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Proof: Let $x\in X$. Let $\varepsilon>0$ be given. Choose $\delta>0$ such that $d(f_{n}(x),f_{n}(y))<\varepsilon$ whenever $d(x,y)<\delta$ and $n\in\mathbb{N}$. Since $H$ is dense in $X$, there exists $y_{0}\in H$ such that $d(x,y_{0})<\delta$. Note that $(f_{n}(y_{0}))_{n}$ is a convergent sequence in $Y$, so there exists $N$ such that $d(f_{m}(y_{0}),f_{n}(y_{0}))<\varepsilon$ whenever $m,n\geq N$.
Now, let $m,n\geq N$ be arbitrary. We have that \begin{eqnarray*} & & d(f_{m}(x),f_{n}(x))\\ & \leq & d(f_{m}(x),f_{m}(y_{0}))+d(f_{m}(y_{0}),f_{n}(y_{0}))+d(f_{n}(y_{0}),f_{n}(x))\\ & < & 3\varepsilon. \end{eqnarray*} Therefore, $(f_{n}(x))$ is a Cauchy sequence in the complete metric space $Y$ and hence $\lim_{n}f_{n}(x)$ exists.