A finite-sample Borel--Cantelli inequality under $m$-dependence

We prove an explicit finite-sample version of the Borel–Cantelli lemma under $m$-dependence. Given any $m$-dependent sequence of events $(A_k{)}_{1\le k\le N}$, we show that $$\mathbb{P}\left(\bigcup _{k=1}^N{A}_k\right)\ge 1-\exp (-\frac{1}{m+1}\sum _{k=1}^N\mathbb{P}(A_k))\mathrm{.}$$ The proof splits the index set into residue classes modulo $m+1$, so that each class consists of mutually independent events, and then applies an elementary product–to–exponential bound. We further derive a quantitative windowed corollary: if the partial sums satisfy ${\sum }_{k=1}^{\phi (n)}\mathbb{P}(A_k)\ge n$ for all $n\ge 1$, then for every $N\ge 1$ and $i\ge 0$, $$\mathbb{P}\left(\bigcup _{k=i+1}^{\phi (i+N)}{A}_k\right)\ge 1-\exp (-\frac{N}{m+1})\mathrm{.}$$ Finally, we present a complementary second-order refinement involving local pairwise intersection probabilities. These results complement the asymptotic and rate results of Lu, Shi and Zhao (2026) by providing explicit finite-$N$ bounds and a simple comparison framework for the baseline and second-order estimates.