A pair of fundamental statements about sequences of events (En) in a probability space: (i) If the sum of probabilities Σ P(En) is finite then the probability that infinitely many of the En occur is zero (first lemma). (ii) If the En are independent and Σ P(En) diverges then infinitely many En occur with probability one (second lemma).

Principle

Summability of event probabilities controls long‑term occurrence: finite sum forces eventual absence almost surely; divergence combined with independence forces persistent occurrence almost surely.

Demonstration

For coin tosses let En be the event that toss n is heads with probability p_n. If Σ p_n < ∞ then with probability one only finitely many heads occur at the specified rare pattern. If p_n are constant positive and events independent, Σ p_n = ∞ and infinitely many occur a.s.

Misapplication

Using the second lemma without an independence (or suitable weak dependence) hypothesis — divergence of Σ P(En) alone does not imply infinitely often in the presence of strong dependence.

Consequence

Gives a simple, widely used criterion for almost sure statements about limsup events; it underpins proofs of almost sure convergence, zero–one laws and probabilistic coverings.

Reversal

The converse of the first lemma fails in general: Σ P(En)=∞ need not imply infinitely often without independence. Dependencies can produce either fewer or more occurrences than suggested by sums alone.

Boundary

Requires a probability space and measurable events; the first lemma holds without independence, the second requires independence or strengthened hypotheses (pairwise independence is not sufficient in general).

Semantic Tension

Tension between deterministic summability criteria and stochastic dependence: the same numeric condition (Σ P(En)=∞) has different implications depending on independence assumptions, so 'Borel–Cantelli' may refer to either lemma and must be disambiguated.

Synthesis

Borel‑Cantelli links the arithmetic of sums of probabilities to almost‑sure occurrence patterns: finiteness of sums forces eventual nonoccurrence a.s., while divergence implies almost sure infinite occurrence under independence, making it a basic tool for turning probabilistic size estimates into long‑run almost‑sure statements.