A typo in Spivak's solution? 
Problem 
Solution


I honestly cannot figure out what he is doing. On one hand, I think Spivak wants to write $|\phi(b)/b^n| > 1/2$ instead of $|\phi(b)/b^2| < 1/2$. On the other hand, how does he justify $b^n(1 + \frac{\phi(b)}{b^n}) > b^n/2$? 
How does he know $\phi(b)/b^n \geq 0$? I guess we could assume $\phi(b) /b^n = 0$ for large $b$, but he doesn't say that at all. 
Finally, I am very perplexed as to why he chooses $b > 0$. In a proof he did in the example, he chose $|x| > 1$. So I first actually thought of doing $|b| > 1, 2\phi(b)$

My guess is that Spivak started by setting the question for the case $n=2$, and then saw that it could easily be generalized, but failed to edit the $2$s to $n$s in a couple of places. It is so easily done, especially with the small characters in subscripts and superscripts! Otherwise the text is correct, as far as I can see. Don't forget that $\phi(b)/b^n$ may be negative; so $|\phi(b)/b^n| < 1/2$ ensures that $1/2<1+\phi(b)/b^n<3/2$ (but we don't need the upper bound). Finally, we need to generate both positive and negative values for $x^n+ \phi (x)$, so $b$ is chosen for the positive value while $a$ is chosen for the negative one.
