The problem:
"True or false: there exists a polynomial $P$ such that $|P(x) - \cos(x)| \leq 10^{-6}$ for all real $x$."
I know the answer (it is false) but I do not understand the proof given in the book.
The proof:
Suppose $P(x) = a_Nx^N + \sum_{n=0}^{N-1}{a_nx^n}$ where $N\geq 1$ and $a_N > 0$. It is enough to show that $P(x) > 2$ for some value of $x$.
(I do not understand the following steps in the proof, could someone please explain them.)
A number $x$ can be found so large that $a_Nx^N > N|a_n|x^n + 2$ for each integer $n$ where $0 \leq n \leq N-1$. The smallest possible value of $P(x)$ for any given positive $x$ would be achieved if all the coefficients in the sum were negative. Thus
$$ P(x) \geq a_Nx^N - \sum_{n=0}^{N-1}{|a_n| x^n} > a_Nx^N + (2 - a_Nx^N) = 2 $$
Link to the book (the problem in question is problem 10): https://books.openbookpublishers.com/10.11647/obp.0181.pdf
