Proof of $\tilde{H}_k(X \lor Y) \cong \tilde{H}_k(X) \oplus \tilde{H}_k(Y)$.

Question

In my algebraic topology course, we showed the following statement: Let $X, Y$ two topological spaces, $(x_0, y_0) \in X \times Y$ and $X\lor Y$ the wedge product of $(X, x_0)$ and $(Y, y_0)$. If $x_0$ is a deformation retract of a neighborhood $U \subset X$ and $y_0$ is a deformation retract of a neighborhood $V \subset Y$, then $$\tilde{H}_k(X \lor Y) \cong \tilde{H}_k(X) \oplus \tilde{H}_k(Y).$$ My teacher told me that the proof was straightforward by considering the Mayer-Vietoris sequence. However I do not really see how we can show this. I thought maybe consider $A = X \cup V$ and $B = U \cup Y$ so that we obtain the exact sequence $$\ldots \to \tilde{H}_k(U \cup V) \to \tilde{H}_k(X\cup V) \oplus \tilde{H}_k(U\cup Y) \to \tilde{H}_k(X \cup Y) \to \tilde{H}_{k-1}(U \cup V) \to \ldots $$ and then use the fact that $U \cup V$ retracts onto the point $[x_0] = [y_0]$ in $X \lor Y$, but I am really struggling with the details. Can anyone help me with this ?