Let P be a polynomial with positive leading coefficient:

$\displaystyle P(x) = a_nx^n +a_{n-1}x^{n-1} + \ldots + a_1x + a_0,$ n>=1

Clearly, as $\displaystyle x\to\infty,\ a_nx^n\to\infty$. Show that, as x --> infinity, $\displaystyle P(x)\to\infty$ by showing that, given any positive number M, there exists a positive number K such that if x>=K, then P(x) >=M.

(Hint: you might wish to choose$\displaystyle K_i>0$ such that

$\displaystyle |a_{n-i}/x^i|< a_n/2n$ when $\displaystyle x > K_i$ and then take the largest of the $\displaystyle K_i$. )