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The complex

$$ \cdots\xrightarrow{}H^k_c(U\cap V)\xrightarrow{j_*}H^k_c(U)\times H^k_c(V)\xrightarrow{i_*}H^k_c(M) \xrightarrow{\mathrm{d}^\ast}H^{k+1}_c(U\cap V)\xrightarrow{j_*}\cdots $$

is an exact sequence

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