I'm reading Achim Klenke's textbook on probability theory, and I've hit a road block in his proof that the strong law of large numbers (SLLN) implies the weak law of large numbers (WLLN).
Let $(X_n)_{n \in \mathbb N}$ be a sequence of real integrable random variables, and let $\widetilde S_n = \sum_{i=1}^n (X_i - \mathbb E[X_i])$, where $\mathbb E[X]$ is the expectation of the random variable $X$. The weak and strong laws of large numbers are:
- WLLN: $\displaystyle \lim_{n \to \infty} \mathbb P\left[ \left| \frac 1 n \widetilde S_n\right| > \epsilon\right]=0$.
- SLLN: $\displaystyle \mathbb P\left[ \limsup_{n \to \infty} \left| \frac 1 n \widetilde S_n\right|=0\right] = 1$.
For $\epsilon > 0$, Klenke defines the sets $A_n^\epsilon = \left\{ \left| \frac 1 n \widetilde S_n \right| > \epsilon\right\}$ and $A = \left\{ \limsup_{n \to \infty} \left| \frac 1 n \widetilde S_n \right| > 0 \right\}$, and states that $$A = \mathop\bigcup_{m \in \mathbb N} \limsup_{n \to \infty} A_n^{1/m},$$ and thus $\mathbb P\left[ \limsup_{n \to \infty} A_n^\epsilon\right] = 0$ for $\epsilon>0$. Then, by Fatou's lemma, \begin{align} \limsup_{n \to \infty} \mathbb P\left[A_n^\epsilon\right] &= 1-\liminf_{n \to \infty} \mathbb E\left[ \mathbb 1_{\left(A_n^\epsilon\right)^c}\right] \leq 1 - \mathbb E\left[\liminf_{n \to \infty} \mathbb 1_{\left(A_n^\epsilon\right)^c}\right] = \mathbb E\left[ \limsup_{n \to \infty} \mathbb 1_{A_n^\epsilon}\right] = 0. \end{align} My question: I understand this entire argument, except for the first equality in the above equation. Equivalently, what I'm trying to figure out is, why is $\limsup \mathbb P\left[A_n^\epsilon\right] = 1-\liminf \mathbb P\left[ \left(A_n^\epsilon\right)^c\right]$? Is this some general measure theory result that I'm not seeing?