= Approximate weakly null lifting through a quotient
If a <Banach space> $X$ has separable dual and $q:X\to X/Y$ is a quotient by a closed subspace, every <weakly null sequence> in the quotient's closed unit ball has a subsequence with approximate weakly null lifts of norm at most $3$. Choose representatives of norm below $3/2$, pass to a common bidual weak-star limit, and subtract <convex blocks> whose quotient images tend to zero in norm. <Convex-block cancellation of a weak-star limit> gives weak nullity and the triangle inequality gives the bound $3$.
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