I'm trying to read through Terence Tao's notes on harmonic analysis and have gotten stuck on one of the basic examples he has provided on the utility of harmonic analysis.
The example aims to use a quantitative estimate on the $L^\infty$ norm of the Fejér sums of a continuous function $f$ to derive the qualitative property of uniform convergence of these sums to $f$.
In particular:
Given a $2\pi$-periodic, continuous function $f$ we have the estimate $||F_nf||_\infty \leq ||f||_\infty$. Where $F_nf$ is the nth Fejér sum of $f$. For trigonometric polynomials, we'll have uniform convergence of these Fejér sums. Additionally, we are able to approximate any continuous function on $[-\pi, \pi]$ via a trigonometric polynomial to any desired accuracy. It follows by the triangle inequality and linearity of the Fejér sums that $F_nf$ converges to $f$ uniformly.
What I can't seem to understand about this proof is the assertion that any continuous function on the interval $[-\pi, \pi]$ can be approximated to any desired accuracy by a trigonometric polynomial. Dr. Tao says this is a result of the Weierstrass Approximation theorem, but I can't see how to extend a statement about approximation by general polynomials to one about trigonometric polynomials. All sources I have found so far use the Fejér sum itself as the approximating polynomial, but that's using the result we're trying to establish. One author hints to being able to use the Weierstrass Approximation theorem directly, stating that we can pass from the interval to the circle, but he doesn't go into any details.
Any help here would be appreciated. Thanks in advance!