I just want to double check some terminology, as it’s not formally been defined, but I’m pretty sure this is what this means. I’ve been given the statement of a theorem:
Weierstrass Approximation Theorem: For $f: [a,b] \to \mathbb{R}$ continuous, $f$ can be uniformly approximated by polynomials.
Presumably by “uniformly approximated by” we mean there exists a sequence $f_1, f_2, \dots$ of (in this case, polynomial) functions such that $f_n \to f$ uniformly? That is,
$$\forall \epsilon > 0 \forall x \in [a,b], \exists N \in \mathbb{N}: \forall n \geq N, |f_n-f| < \epsilon.$$
Is this correct?