Proof:

The long exact sequence for a pair and its naturality gives a commutative diagram with exact rows.

$$\begin{CD} @> >> H_n(U\cap U) @> >> H_n(V) @> >> H_n(V,\; U\cap V... ...) @> >>H_n(U\cup V) @> >> H_n(U\cap V,\; U) @> >> H_{n-1}(U) @> >> \\ \end{CD}$$

Applying the excision theorem to the inclusion $(U\cap V,\;V)\longrightarrow (U\cup V,\; U)$, we see that the third arrow is an isomorphism in the displayed diagram. The result now follows from lemma (38.3).