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Let $p:\mathbb{R}\rightarrow\mathbb{R}$ be a polynomial of odd degree. Prove that there is a solution of the equation $$p(x)=0, x\in\mathbb{R}$$

I am giving this question in an analysis textbook and the only machinery I have to work with is the sequential definition of continuity and the intermediate value theorem. Using only these ideas I am having difficulty coming up with a concrete proof.

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  • $\begingroup$ $\lim_{x\to+\infty} p(x) = -\lim_{x\to-\infty} p(x) = \pm\infty$. So we can pick $x$ sufficiently large that $p(x) < 0$ and $p(-x) > 0$. Then apply the IVT. $\endgroup$ – Xander Henderson Apr 12 '18 at 23:37
  • $\begingroup$ What I am missing is a concrete proof that those limits are as stated using the sequential definition of convergence or divergence of a function. $\endgroup$ – Walt Apr 13 '18 at 2:19
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Hint: if $p$ is a polynomial of odd degree, then $$\lim_{x\to +\infty}p(x)=\pm \infty, \quad \lim_{x\to -\infty}p(x)=\mp\infty $$

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  • $\begingroup$ The problem as I have stated it is that I need a concrete proof using sequences. Essentially if given a sequence call it $\{a_n\}$ of real values which diverges to infinity I have to show that $p(a_n)$ diverges as well for any odd degree polynomial. $\endgroup$ – Walt Apr 13 '18 at 2:19

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