How to show the failure of the homotopy extension property for the pair $I=[0,1]$ and $A=\left\{ 0,1,\frac{1}{2},\frac{1}{3}...\right\}$?

Question

I'm facing difficulties regarding a statement which Hatcher makes in page 14 of his 'Algebraic Topology' book—that there's no continuous retraction from $I\times I\rightarrow I\times\left\{0\right\}\bigcup A\times I$ where $I$ is the unit interval and $A=\left\{0,1,\frac{1}{2},\frac{1}{3}...\right\}$. How does one rigorously show this? Thanks in advance.