Wedge sum of contractible spaces Let $\vee_i U_i$ be the wedge sum of a set of based spaces $(U_i,x_i)$. Suppose each $U_i$ is a contractible neighborhood of $x_i$, then I want to prove that $\vee_i U_i$ is also contractible. (J. P. May mentions this statement in Section 2.8 of his book A Concise Course in Algebraic Topology)
My attempt: Suppose $F_i: c_{x_i} \simeq id_{U_i}$ are homotopies, where $c_{x_i}: U_i \to U_i$ is the constant map with value $x_i$. I tried to construct a homotpy $F$ between $c_x:\vee_i U_i \to \vee_i U_i$ and $id_{\vee_i U_i}$ via $F|_{U_i}=F_i$, where $x$ is the homotopy class of $x_i$. But the problem is that for fixed $t \in [0,1]$, $F_i(x_i, t)$ are different in general, so $F$ is not well-defined at $x$.
Q: How to explicitly contruct a homotopy between $c_x$ and $id_{\vee_i U_i}$?

The statement is wrong, the Griffiths Twin Cone is a counterexample, as suggested by JHK in the comment above. So when May says $U_i$ is a contractible neighborhood of $x_i$, I guess he means $x_i$ is a strong deformation retraction of its open neighborhood $U_i$.
