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 $|b| > 1, 2\phi(b)$

He is correct except for the $2$ exponent. He can choose such a $b$ because $\lim\limits_{x\to\infty}\phi(x)/x^n=0$. The other inequality follows from the "reverse triangle inequality" $|u+v|\ge |u|-|v|$.
