I read and understood the following definition*:
Define $h(x)=|x|$ on the interval $[-1,1]$ and extend the definition of $h$ to all of $\mathbb{R}$ by requiring that $h(x+2)=h(x)$. The result is a periodic "sawtooth" function.
However, I was then promptly asked to sketch the graph of $(1/2)h(2x)$ on $[-2,3]$ and to give a qualitative description of the functions $h_n(x)=\frac{1}{2^n}h(2^nx)$ as $n$ gets larger.
However, this is where I am confused: Is it not the case that $h_n(x)=\frac{1}{2^n}h(2^nx)=\frac{1}{2^n}|2^nx|=\frac{2^n}{2^n}|x|=|x|,$ and $h_n$ behaves just like $h$ regardless of the size of $n$? Either I am misunderstanding something or the book has made a typo. Could anyone clarify this for me? Thanks in advance!
*This is from Stephen Abbott's Understanding Analysis.