Morse inequalities (${\sum }_{i=0}^j(-1{)}^{j-i}(c_i-b_i)\ge 0$)

For a closed manifold and a Morse function, the displayed strong inequalities hold for every $j$, with equality at the top dimension. Indeed the Morse chain complex rank formula makes the alternating sum equal to $r_{j+1}\ge 0$. Adding the inequalities at $j$ and $j-1$ gives $c_j\ge b_j$. The weak inequalities, even together with equality of Euler characteristics, are less restrictive: $b=(1,0,0,1)$ and $c=(2,0,0,2)$ satisfy those numerical conditions but violate the strong inequality at $j=1$.