Can Polynomials Approximate a Bounded Function on the Entire Real Line?

According to the Stone-Weierstrass theorem, a polynomial can approximate a continuous function on a closed interval.

I agree that, a polynomial cannot approximate a function that is unbounded on an open interval because the polynomial itself would be bounded on that interval, while the target function is not. This is discussed in detail here.

My question: Can a polynomial approximate a function that is bounded on the entire real line, $(-\infty, +\infty)$?

For example, consider the cumulative distribution function of the normal distribution $\Phi$, which is bounded by 0 and 1 on the real line.Can it be approximated by a polynomial?