So, one needs to prove that every regular Lindelöf space is normal, exactly as the title suggests.
The Attempt at a Solution
I used the following theorem:
Every regular space with a countable basis is normal.
Now, what we need to prove can be proved very similarily to the proof of the theorem above. It's Theorem 32.1., page 200, in Munkres.
What I had in mind:
The proof is exactly the same, with one variation.
Let B be a basis for X. We choose a basis element contained in V for every x in A. Now, for any x in X\A, choose a basis element containing X. This collection forms an open cover for X, and since X is Lindelöf, it has a countable subcollection. So, the subcollection of all the basis elements for the elements of A is countable. Hence, the rest of the proof is the same.
I hope it won't be a problem to open Munkres and look at the proof, since it was too long to type, so I decided to be practical.
Thanks in advance.