Sequential repetition amplifies completeness and soundness gaps

For the honest prover, the acceptance indicators are independent Bernoulli variables of mean at least $c$; the lower-tail Chernoff bound A Chernoff bound for sums of independent Bernoulli trials, applied to these success indicators, makes majority failure exponentially small in the number $k$ of copies.

Against any adaptive prover, conditional on every earlier transcript, the next copy accepts with probability at most $s$ by Completeness and soundness of an interactive protocol. Induction on copies gives $\mathbb{E}{e}^{\lambda \sum X_i}\le (1-s+s{e}^{\lambda }{)}^k$, so the usual Chernoff upper-tail calculation makes majority acceptance exponentially small.

Choose odd $k$ large enough that both exponential bounds are at most $\epsilon$; for fixed $c,s,\epsilon$ this is constant and preserves polynomial bounds.