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To prove this, use naturality of the Leray-Serre spectral sequence: the $G$-equivariant map $\ast \to M$ induces a map of spectral sequences $$ H^p(G;H^q(M;\Bbb Z)) \to H^p(G;H^q(\ast;\Bbb Z)). $$ This map is compatible with the differentials, the map is an isomorphism when $p=0$, and the target is zero when $p > 0$. If an element $y$ on the 0-line were in the image of a differential, $y = d_r(x)$, then applying this map of spectral sequences we'd find $y = d_r(0)$ and hence $y=0$.