Let $X$ be a topological space and $A \subseteq X$ a subspace, and let $a_0 \in A$. Suppose that there exist $H:X \times I \to X$ continuous, such that $H(x,0)=x$ for every $x\in X$, $H(A \times I) \subseteq A$ and $H(a,1)=a_0$ for every $a\in A$. Show that $q: X \to X/A$ is a homotopy equivalence.
So I have some ideas in mind but im not being able to put them together.
Because of the hypothesis given for $H$, we have that $H|_{A\times I}:A \times I \to A$ is a contraction, and therefore $A$ is conctractible, which implies it has the same homotopy type than a one point space $\{p\}$. So I think that $X/A$ and $X/ \{p\}$ should be homotopically equivalent (im not sure about this, i wasn't able to prove it, but its what my intuition suggests) and if this is true, then (i think) proving that $q_1 : X \to X/\{p\}$ is a homotopy equivalence should be easier.
However, Im not being able to write formally any of this ideas. Are there even true?