# Restriction of cellular Homotopy equivalence

I'm trying to solve a problem of Hatcher's Algebraic Topology:

"Use cellular approximation to show that the n-skeletons of homotopy equivalent CW-complexes without cells of dimension n+1 are also homotopy equivalent."

Using cellular approximation I can get a cellular map which is also a homotopy equivalence between the two CW-complexes. Should I prove that the restriction of it to the n-skeleton is still a homotopy equivalence? Is there any theorem about how to get a homotopy equivalence after a restriction?

This is what I did so far:

Let $$f:X\to Y$$ be the homotopy equivalence and $$\widetilde{f}$$ its homotopic cellular map with $$H:X\times [0,1]\to Y$$ the homotopy between them. I know that $$X\times [0,1]$$ inherits a CW-structure from $$X$$ and therefore, I can use again cellular approximation to get a cellular homotopy $$\widetilde{H}$$, homotopic to $$H$$. By assuming $$X_m=X_{m+1}$$ and $$Y_m=Y_{m+1}$$, I get $$(X\times [0,1])_{m+1}=\bigcup_{p+q=m+1}X_p\times [0,1]_q= X_{m+1}\times \{0\}\cup X_{m+1}\times \{1\} \cup X_{m}\times [0,1]= X_{m}\times [0,1]$$ then the restriction of $$\widetilde{H}$$ to $$(X\times [0,1])_{m+1}$$ is still a homotopy between two maps $$X_m\to Y_m$$. Hence I have two maps that in some way are homotopic to $$f$$. Am I going somewhere?

You have maps $$f : X \to Y$$ and $$g : Y \to X$$ which are homotopy inverse to each other. Choose cellular maps $$f' : X \to Y$$ and $$g' : Y \to X$$ homotopic to $$f$$ and $$g$$, respectively. Then they are also homotopy inverse to each other. We claim that the restrictions $$f_n' : X^n \to Y^n$$ and $$g_n' : Y^n \to X^n$$ of $$f'$$ and $$g'$$ are homotopy inverse to each other.
There exists a homotopy $$H : X \times I \to X$$ from $$g' f'$$ to $$id$$. Its restriction to $$X \times \{0,1\}$$ is cellular, thus it is homotopic rel. $$X \times \{0,1\}$$ to a cellular map $$H' : X \times I \to X$$. We have $$H'_0 = g' f'$$ and $$H'_1 = id$$. Since $$X^n \times I \subset (X \times I)^{n+1}$$ and $$X^{n+1} = X^n$$, we get $$H'(X^n \times I) \subset X^{n+1} = X_n .$$ This shows that $$g'_n f'_n \simeq id$$. That $$f'_n g'_n \simeq id$$ is similar.