To be clear, this is the theorem of interest:
Theorem – Banach-Saks: Let $(X,\mathcal{A},\mu)$ be a measure space and $1\lt p\lt\infty$. If $\{f_n\}_{n\in\Bbb N}\subset\mathcal{L}^p(X)$ and $f_n\rightharpoonup f$ in $\mathcal{L}^p(X)$, then there is a subsequence $\{f_{n_k}\}_{k\in\Bbb N}\subset\{f_n\}_{n\in\Bbb N}$ whose Cesaro mean converges strongly in $\mathcal{L}^p$ to $f$.
Very annoyingly, Royden's real analysis proves this theorem only for $p=2$ and cites another source for the proof with more general $p$. I have been unable to find such a source; all I had found was a sparse proof sketch that a university had posted as homework. I am not a university student however, and as I was struggling with this proof sketch I discovered that the website had gone down. Perhaps this is a temporary issue, but my only resource on this theorem is now lost!
All attempts to google "Banach Saks theorem proof" have resulted in academic papers about the "Banach Saks property", i.e. these papers were proving other things with this Banach Saks theorem as an assumption.
From what I could gather from the proof sketch I saw, the details are messy and non trivial for someone with my education to come up with on their own – and that was only the sketch! If anyone knows perhaps a straightforward proof, or a link to a proof, I'd greatly appreciate it! I don't mind if the linked proof is hard or dense, just so long as it's mostly complete. Even hints on proving it for myself would be appreciated; for the reference ability level, I've followed Royden's book (and other measure theory resources) just fine up until this point.
Links are left for other students such as myself; I believe it is to the user Will Jagy on this site that I owe the pleasure of having Royden's book in the first place!
Best Answer
A proof of the theorem for $p>1$ can be found in Théorème 1 in the original paper by Banach and Saks:
Banach, Stefan, and Stanislaw Saks. "Sur la convergence forte dans les champs $ L^{p}$." Studia Mathematica 2.1 (1930): 51-57.
For a proof in the case $p=1$, you may see
Szlenik, W. "Sur les suites faiblement convergentes dans l'espace L." Studia Mathematica 25 (1965): 337-341.