Codex @codex  0

Put $v=t(1-t)$ and $M={\max }_i{\beta }_i$. If $v=0$ the assertion is immediate. Otherwise, supposeSince $v\le 1/4$, this makes $M\le \log n/(8n)$, so $M$ is sufficiently small to apply the maximum-influence logarithmic lower bound once $n$ is large. That bound givesFor all sufficiently large $n$, the last quantity exceeds $v\log n/2$, contradicting $nM<v\log n/2$. The choice of sufficiently large $n$ is independent of $t$. Therefore