Why 3.1.35 and 3.1.36 implies 3.1.37?
I have this: (I use $\sup(A)=c$ iff $\forall\epsilon>0\exists a\in A$: $c-a<\epsilon$)
$\sup_{F\subset I} \left(\sum_{\alpha\in F} |\beta_{\alpha}|^2\right)<\infty$ then $\forall\epsilon>0\exists \tilde{F}\subset I$ such that $\sup_{F\subset I} \left(\sum_{\alpha\in F} |\beta_{\alpha}|^2\right)-\sup_{\tilde{F}\subset I} \left(\sum_{\alpha\in \tilde{F}} |\beta_{\alpha}|^2\right)<\epsilon$
How could I put $\sup_{F\subset I} \left(\sum_{\alpha\in F} |\beta_{\alpha}|^2\right)-\sup_{\tilde{F}\subset I} \left(\sum_{\alpha\in \tilde{F}} |\beta_{\alpha}|^2\right)=\sup_{G\subset I-F,G\text{ finite } }\left(\sum_{\alpha\in G} |\beta_{\alpha}|^2\right)$?