The proof of the homotopy extension and lifting property (HELP)

Question

I am reading J. P. May's Book, the section about homotopy extension and lifting property (HELP) on page 75:

I know this is true for $$(X,A)=(D^n,S^{n-1})\cong (CS^{n-1},S^{n-1}),$$ where $CS^{n-1}$ is the unreduced cone $S^{n-1}\times I/S^{n-1}\times \{1\}$.

But I have no idea about how to prove for general CW pair. I tried to use induction by extending the diagram on the left side with the push out diagram $S^{n-1}\to D^{n}$ and $S^{n-1}\to X^{n}$ where $X^n$ is the $n$-skeleton of $X$. But it did not work.

Could anyone give more details that how this induction works? Thanks!

Answer 1

See Lemma 2.3 in this notes by Agnès Beaudry.