How can one prove that manifolds are regular?

How can one prove that manifolds are regular?

First, some clarification of the definition of a manifold that I'm using:
A manifold $M$ is a Hausdorff, locally Euclidean and second countable
topological space.
Now, I am trying to prove that Manifolds are paracompact, and I have
established most of the details for the proof from the link above after
getting so far under my own steam. The only hole I have, however, is the
assertion that manifolds are regular. From what I can infer, this comes
from the properties of being locally path-connected and Hausdorff, but I
cannot make the leap from those two properties to the required regularity
to complete the proof.
Apologies for the probably quite elementary question; I'm an applied
mathematician, and am aiming to be well-read in a range of mathematical
topics for my own interest, and while I have a textbook I'm working
through on the topic of differentiable manifolds, there's still a certain
unfamiliarity with the methods employed in certain pure maths topics.
Perhaps even a hint would be best, as I really do like to attempt to grasp
these things by myself as much as possible, but I really just can't seem
to get this result out... Thank you in advance for whatever help you can
give me.