Does there exist a sequence of polynomials converging uniformly to $f(x) = |x| $ on $ℝ$?

Question

Does there exist a sequence of polynomials converging uniformly to $f(x) = |x|$ on $ℝ$?

I understand this is possible on a closed interval due to Weierstrass Approximation Theorem.

Intuitively, I think there's no such sequence on $ℝ$ but I'm not sure how to prove this rigorously.

Any hints would be so useful!

Answer 1

There are not any $p_1 \neq p_2$ such that $|p_1 -p_2|$ is bounded, unless $p_1 - p_2$ is a constant. This means that for a sequence of polynomials to converge uniformly, all the coefficients but the constant terms must be (eventually) fixed, so the limit of such a sequence must be a polynomial. $|x|$ is not a polynomial.

Answer 2

That is not possible uniformly on $\mathbb R$. Clearly, it is not possible with a sequence of polynomials of degree less than $N$, whatever $N$ you chose. Therefore degrees of the approximating sequence MUST explode. But polynomial function grow so much faster at infinity than $|x|$ that uniform convergence is impossible.