Statement: There exist a polynomial $P$ such that $|P(x) - \cos(x)| \leq 10^{-6}$ for all (real) $x$.
My answer: False. All polynomials of a degree $n \geq 1$ are unbounded as $x$ tends to infinity. A polynomial of degree $n = 0$ is bounded only when it is in the form $y = a$ (horizontal line) but this will not help because $\cos(x)$ varies between $[-1,1]$.
My question: Is my answer reasonable? I am particularly concerned about 'unbound\ed' polynomials as this is the term I just made up.
Please don't give me complete answer, I just want to know the flaws in my argument and get some hints to better solutions.