Let $A\subset\mathbb R^n$ be star-convex, that is there exists $a_0\in A$ such that for each $a\in A$, the line segment from $a_0$ to $a$ is contained in $A$. Define $H:A\times[0,1]\to\mathbb R^n$ by $H(a,t)=(1-t)a + ta_0$.
The book (Dieck's Algebraic Topology) claims that $H$ is a null homotopy of the identity, and hence star-convex sets are contractible.
I see that a null homotopy of a map $f$ is a homotopy between $f$ and a constant map, and that a null homotopy of the identity is a contraction of the underlying space. However, I do not see how the homotopy $H$ defined here is a null homotopy, or that it is a null homotopy of the identity. Can anyone help me to understand this?