Proof of Borel-Cantelli Lemmas

Preliminaries. We use two facts about any measure $\mu$. Countable subadditivity: for events $(B_m)$, $\mu(\bigcup_m B_m)\le\sum_m\mu(B_m)$, by disjointifying ($D_m=B_m\setminus\bigcup_{l<m}B_l$), countable additivity, and monotonicity ($\mu(D_m)\le\mu(B_m)$, since $\mu(B_m)=\mu(D_m)+\mu(B_m\setminus D_m)$ by finite additivity). Continuity from below was established in Step 0 of the proof of Monotone Convergence Theorem; applying it to complements gives continuity from above for probability measures: if $C_1\supseteq C_2\supseteq\cdots$ are events then $P(\bigcap_k C_k)=\inf_k P(C_k)$, since the complements increase to the complement of the intersection and $P$ is finite.

First lemma. For every $k$, monotonicity and countable subadditivity give

and the right side tends to $0$ as $k\to\infty$, being the tail of a convergent series (its partial sums are bounded by the total sum minus the $(k-1)$st partial sum). Hence $P(\limsup_m A_m)=0$.

Second lemma. First, complements of independent events are independent: replacing one event $A_i$ by its complement $\Omega\setminus A_i$ preserves the product formula, since for any nonempty index set $S$ containing $i$,

by finite additivity and the product formula; iterating handles any number of complementations.

Next, $1-x\le\exp(-x)$ for every $x\ge 0$: the function $h(x)=\exp(-x)-(1-x)$ satisfies $h(0)=0$, and by claim 3 of Basic Properties of the Exponential Function together with the chain rule its derivative is $h'(x)=1-\exp(-x)$, which is $\ge 0$ for $x\ge 0$ because $\exp(x)\ge 1$ there (claim 4) and hence $\exp(-x)=1/\exp(x)\le 1$ (claim 2); by the mean value theorem, $h$ is nondecreasing on $[0,\infty)$, so $h(x)\ge 0$.

Fix $k$ and $K\ge k$. By independence of the complements, the product formula, the bound just proved, and the multiplicativity of $\exp$ (claim 1 of Basic Properties of the Exponential Function),

Since $\sum_m P(A_m)=\infty$, the exponent tends to $-\infty$ as $K\to\infty$, so by claim 4 of Basic Properties of the Exponential Function the right side tends to $0$; by continuity from above (the intersections decrease in $K$),

The events $\bigcup_{m\ge k}A_m$ decrease in $k$, so by continuity from above once more,