Let $A\in\mathcal{L}(X,Y)$ be some bounded linear operator from $D$ to $Y$, and $D\subset X$ be a dense subspace. Let $B$ be defined so that $B(x)=y=\lim_{n \rightarrow \infty}Ax_n$ for any sequence $\{x_n\}_n$ in $D$ converging to $x$. We know that $\{Ax_n\}_n$ converges and the limit $y$ is unique.
Prove that the function $B$ satisfies $B(x)=Ax$ for all $x \in D$.
I figured defining a constant infinite series, $xn=x$ for all $n$ would be a start, but I'm not sure where to go from here. This is a course for introduction to real analysis, so any help is much appreciated