Show that a Regular space $X$ is Lindelöf if it has a locally finite open cover $\mathcal{U}$ such that $\rm{Fr}\,U$ has the Lindelöf property

Question

Show that if a regular space $X$ has a locally finite open cover $\mathcal{U}$ such that $\rm{Fr}\,U$ has the Lindelöf property for all $U\in\mathcal{U}$, then the space X is also a Lindelöf Space.

one suggestion for this exercise please?. I try of all but i don't have exit.

Thanks for your help.