= Gliding-hump continuity principle
{title2=$U_nSP_n\text{ is continuous eventually},\quad P_n=T_1\cdots T_n$}
Let $S:E\to Y$ be linear between <Banach spaces>, let $T_n:E\to E$ be bounded, and let $U_n:Y\to Z_n$ be bounded. If $U_nSP_m$ is <continuous> whenever $m>n$, then $U_nSP_n$ is <continuous> for all sufficiently large $n$. After rescaling the <bounded operators>, assume their <norms> are at most one. If the conclusion failed, discontinuity would allow successively tiny <vectors> $x_j$ whose diagonal images dominate all previous contributions. Choose their <norms> also to control the <continuous> maps $U_{n_i}SP_{n_i+1}$ on subsequent tails. The convergent sum $x=\sum_jP_{n_j}x_j$ would then satisfy $\|U_{n_j}Sx\|>j$ despite $\|U_{n_j}Sx\|\le\|Sx\|$, a contradiction. The tail is factored through $P_{n_i+1}$ before applying $S$, so no <continuity> of $S$ is assumed.
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