I'm stuck on this proof for a problem set and would appreciate any pointers in the right direction: Prove that if $f$ is a $B$-valued function, and if $E \in S$ ($S$ is the sigma algebra)with $\mu(E) < \infty$, then for any $\epsilon > 0$ there is an $F \subseteq E$ and a constant $K$ such that $\mu(E \setminus F)< \epsilon$ and $\lVert(f(x))\rVert < K$.
I understand how to satisfy the conditions for the set $F$ as it just follows from Egoroff's theorem, but I'm not sure how to show that $\lVert(f(x))\rVert$ is bounded on $F$. We haven't gone over Lusin's theorem yet, so I can't use that.
EDIT By $B$ valued it is meant valued in the Banach space.
$\mu \{x\in E: \|f(x)\| \geq K\} \to 0$ as $K \to \infty$ (because the sets here decrese to empty set and $\mu (E)<\infty$) so there exists $K$ such that $\mu \{x\in E: \|f(x)\| \geq K\} <\epsilon$. Take $F=\{x\in E: \|f(x)\| < K\}$.