Convex block
= Convex block
{title2=$u_n=\sum_{i=p_n}^{q_n}a_ix_i$}
A convex block of a sequence in a <Banach space> is a sequence of <convex combinations> supported on successively disjoint finite intervals, with $p_n<q_n<p_{n+1}$, $a_i\ge0$, and $\sum_{i=p_n}^{q_n}a_i=1$. Zero coefficients can fill gaps and ensure strict endpoint inequalities. By the <Mazur theorem>, every <weakly null sequence> has convex blocks tending to zero in norm.