
Prove P(x) >= M
Let P be a polynomial with positive leading coefficient:
$\displaystyle P(x) = a_nx^n +a_{n1}x^{n1} + \ldots + a_1x + a_0,$ 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
$\displaystyle a_{ni}/x^i< a_n/2n$ when $\displaystyle x > K_i$ and then take the largest of the $\displaystyle K_i$. )