# Best Approximation Theorem - Fourier Series

I'm reading Stein & Shakarchi's Fourier Analysis, but had a question about a small part of the proof of the mean-square convergence of fourier series of a function $$f$$.

To show the convergence, they first proved the Best Approximation Lemma, saying that fourier series is the best trigonometric polynomial estimating f for degree at most N. Then on page 79, they used a corollary 5.4 in Chapter 2 saying that there exists a polynomial $$P$$ of degree $$M$$ s.t. $$|f(\theta)-P(\theta)|<\epsilon$$ for all $$\theta$$. This is basically the contents of that corollary, and I'm totally okay with that. After some algebra, and using best approximation theorem, we can certainly get $$||f(\theta)-S_M(f)||<\epsilon$$ Note that the best approximation theorem has both sides to have the same degree. How can we conclude that $$||f(\theta)-S_N(f)||<\epsilon,\ \forall N \geq M\ ?$$

Why is it true for $$N > M$$ as well?

Definition: 1. $$||f(\theta)-S_M(f)||^2 = \frac{1}{2\pi}\int_{0}^{2\pi}|f(\theta)-S_M(f)|^2 d\theta$$. 2. $$S_N(f) = \sum_{-N}^{+N} a_ne^{in\theta}$$ is the $$Nth$$ partial sum of $$f$$, where $$a_n$$ is the $$Nth$$ fourier coefficient of $$f$$.

Thanks in advance for any insights.

So, then, we have by other means a trig polynomial of degree $$M$$ that approximates to within $$\epsilon$$. By the lemma, for all $$N\ge M$$, the $$N$$th Fourier partial sum is at least as good as everything else with degree $$\le N$$ - including that degree-$$M$$ approximation we found.