If a sequence of polynomials converges uniformly to a continuous function on the real line, then this function is a polynomial

Asked Dec 14 '18 at 18:56

Active Dec 14 '18 at 18:56

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I'm trying to prove that if $f\colon\mathbb{R}\to\mathbb{R}$ is continuous function and there is a sequence of polynomials $p_n$ that converges uniformly to $f$ on $\mathbb{R}$, then $f$ is a polynomial itself.

It smells like somewhat near the Weierstrass approximation with polynomials, but I have no idea how to use uniform convergence on the entire real line, since this statement is wrong on any bounded interval.

Any hint or solution will be appreciated.

asked Dec 14 '18 at 18:56

Hasek

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I have a feeling it has something to do with non-constant polynomials diverging off to infinity. If two polynomials have a different degree, then their distance (which I’d define here as the sup over $x \in\mathbb{R}$ of the distance at $x$) must be infinite. – SvanN Dec 14 '18 at 19:01

If a sequence of polynomials converges uniformly to a continuous function on the real line, then this function is a polynomial

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