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The complex

$$ \cdots\xrightarrow{}H^k(M)\xrightarrow{i^*}H^k(U)\times H^k(V)\xrightarrow{j^*}H^k(U\cap V) \xrightarrow{\mathrm{d}^\ast}H^{k+1}(M)\xrightarrow{i^*}\cdots $$

is an exact sequence

Just apply the Connecting Homomorphism Theorem to the Mayer-Vietoris short exact sequence of complexes