I have some question so this passage.
(1) Are the $a_i$ the critical values of the function $f$, meaning that $f(p_i)=a_i$, where the $p_i$ are the critical points of $f$?
(2) Why can we assume that the $a_i$ are in increasing order? This makes sense to me if $f$ is some sort of height function. But I don't see how we can assume this for an arbitrary Morse function.
1) The $a_i$ are exactly chosen to be non-critical values. Assume all the critical points $p_i$ have different critical values $c_i$. Then you can choose the $a_1$ to be between $c_i$ and $c_{i+1}$ (where $a_k$ is anything bigger than the maximum value of the function).
2) You can wiggle a Morse function a bit so that all critical points assume different critical values. This will not change the index of the critical points, nor the number of them. Hence if the Morse inequality holds for functions all of whose critical points have distinct critical values, you can also prove it for any Morse function. Is there a Lemma in Milnor's book that proves this?