I'd like to show that the function $p: \mathbb{R} \rightarrow \mathbb{R}$, $p(t) = a_0 + a_1 t+ \cdots a_n t^n $ ($a_n \neq 0$) is unbounded (it means $\forall M>0$, $\exists x\in \mathbb{R}$ such that $|p(x))| > M$) without using limits.

I wrote $$|p(t)| = |t^n| \bigg| \frac{a_0}{t^n} +\frac{a_1}{t^{n-1}}+ \cdots + a_n \bigg| $$

And tried something like $|t|>a$ but I had trouble with $\frac{a_0}{t^n}, \frac{a_1}{t^{n-1}},\cdots$.

What is a good inequality to start with?

Thank you.

  • 1
    $\begingroup$ One way to do this is use $|x+y|\geq |x|-|y|$. $\endgroup$ – Matt Samuel Dec 13 '15 at 23:36

for a polynomial function $g(x)$ suppose for all $M \gt 0$ we can find $r_M \gt 0$ such that $$ x \gt r_M \Leftrightarrow |g(x)| \gt M $$ let us call such a function eventually large, which may be abbreviated to $EL$

clearly $x$ is $EL$ and we have:

(1) if $a$ is a constant and $g$ is $EL$ we have $a+g$ is $EL$

(2) if $g$ is $EL$, then so is $xg$

hence any polynomial is $EL$


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