I'm self learning Rotman's Algebraic Topology and I've come across the following problem:
I'm trying to prove Theorem $9.12(i)$ (attached in first image below) inductively on $k$, but I'm having trouble understanding how to continue.
I see that if $k=0$ then the theorem is seen to be true by Lemma $9.11$ (attached in the bottom image below) by connecting the $0$'s with a $0$ map, but I'm not sure how to show how the other cases are true inductively on $n \ge 0$.
Anyone have any ideas? Clear and detailed explanations would be helpful as I am self-learning this.
