Differential Geometry – A question on Atiyah's proof of the convexity of the moment map

It's about proving the connectivity of the level sets of the moment map $ mathbb T ^ n $ $ implies $ convexity of the image of the moment map for $ mathbb T ^ {n + 1} $ action. I am the wonderful lecture notes of Ana Cannas, but I found myself stuck at one point of the proof where it is said for two points $ p_0, p_1 in M ​​$ there is a sequence of points $ q_n rightarrow p_0, l_n rightarrow p_1 $ such as $ mu (p_1) – mu (p_0) $ is in $ ker A ^ t $ for a whole number $ (n + 1) times (n) $ matrix $ A $ with rank n.

I realized that it would be enough to get a sequence such that $ mu (p_1) – mu (p_0) $ all the rational coefficients are enough, but I'm confused on how to do it.

I've been watching this part of Dusa Mcduff's book and she's not developing that topic either, so I have the impression that it may be really trivial, but I guess that I miss a hint / an easy observation.