Show that if $\varphi: X \rightarrow Y$ is a homotopy equivalence, then the induced homomorphisms $\varphi_{*}:\pi_n(X,x_0) \rightarrow\pi_n(Y,\varphi(x_0))$ are isomorphisms, for all n$\in \mathbb{N}$.
I've already checked the proof given in 1.18 (case n=1) but I can't generalize to arbitrary dimension. Intuitively, I've tried to repeat the argument, in this case using the change-of-basepoint isomorphism given in page 341, but I can't write a correct proof. I'd appreciate if someone could help me with it. Thanks.
Here is one other attemp: If $\varphi: X \rightarrow Y$ is a homotopy equivalence, $\exists \psi:Y\rightarrow X$ s.t. $\varphi \circ \psi \simeq Id_Y$ and $\psi \circ \varphi \simeq Id_X$, so the induced applications statisfies:
$$(\varphi \circ \psi)_{*}\stackrel{Functor}{=}\varphi_{*} \circ \psi_{*}=Id_{\pi_n(Y)}$$ $$(\psi \circ \varphi)_{*}\stackrel{Functor}{=}\psi_{*} \circ \varphi_{*}=Id_{\pi_n(X)}$$ so $\varphi_{*}^{-1}=\psi_{*}$ and $\varphi_{*}$ is an isomorphism of groups.
Use $\psi:Y\to X$ such that $\psi\circ\varphi\simeq id_X$ and $\varphi\circ\psi\simeq id_Y$. What about $\psi_*\circ \varphi_*$ and $\varphi_*\circ\psi_*$