Eq. (89), p. 21 (proof of Lemma 19) — Chernoff bound: 2^{−2b} Σ_{i=j+1}^{b} C(2b, b+i) ≤ e^{−j²/b}

For all integers $b, j \ge 0$, the probability of seeing more than $b+j$ heads on flipping $2b$ fair coins is at most $e^{-j^2/b}$:

This is the bound on the coefficients of the Chebyshev series (77) that makes its tail beyond $j_0$ negligible in Lemma 19.

Formalization Note No restriction on $b$ or $j$ is imposed, matching the unrestricted display. At $b = 0$ the left side is the empty sum $0$ and Lean reads $-j^2/0$ as $0$, so the right side is $1$; for $j \ge b$ the left side is the empty sum $0$. Both corners are true and carry no content.