Let $f(x)$ be a smooth, bounded real function, and consider the closed curve that is $f$ on the interval $[x_1,x_2]$.
Let $g_n(x) = \sum_{k = 0}^{n} a_{n,k} x^k$ be the best $n$th order polynomial least-squares fit to $f$ on the given interval.
Given some $\delta > 0$, it seems evident to me that there always exists an $n$ such that $$\left\| f(x) - g_n(x)\right\|_2 \leq \delta$$ where $\left\|\cdot\right\|_2$ denotes the $L^2$ norm over the interval $[x_1,x_2]$, no matter what $f$ is, so long as it's smooth and bounded.
Hence, we can make the claim that "on any finite interval, $f$ can be approximated to arbitrarily high precision by a polynomial" and ultimately, "we can safely substitute $g_n$ for $f$ in proofs involving $f$ since $\mathop {\lim }\limits_{n \to \infty} \left\| f(x) - g_n(x)\right\|_2 = 0$".
Is my reasoning here sound?
If not, what would be an example of an $f$ that couldn't be approximated in this way?
If so, is there a formal name for this ability of $f$ to always be representable by a polynomial (i.e. power series) over a finite interval?
Here's what you may be looking for: Stone Weierstrass