Let $X$ and $Y$ are BS and $F\in BL(X,Y)$ surjective. Let $y_n\to y$ in $Y$ and if $F(x)=y$ then there is a sequ $x_n$ s.t $F(x_n)=y_n$ for all n.

Question

The question asks to find a sequence $x_n$ in $X$ (Banach space) such that $F(x_n)=y_n$ for all $n$ and $x_n\to x$ in $X$. Where it is given that $y_n\to y$ in $Y$ and $F$ is surjective.

From open mapping theorem we know that $F$ is an open map. But I am not getting how it os going to help in getting a sequence $x_n$ in $X$ such that $F(x_n)=y_n$ for all $n$ and $x_n\to x$ in $X$.