Proof of Weak Law of Large Numbers

Fix a real $\varepsilon>0$, and for $n\in\mathbb{N}$ write $S_{n}=\sum_{k=1}^{n}X_{k}$.

Mean and variance of $S_{n}$. Let $n\in\mathbb{N}$. Each $X_{k}$ is integrable (Lᵖ Inclusions and Constants on a Finite Measure Space (i)), so by linearity (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (viii), with all coefficients equal to $1$) and by Expectations and Variances of Independent Random Variables (iii), $S_{n}\in\mathcal{L}^{2}(P)$ and

$$\mathbb{E}S_{n}=\sum_{k=1}^{n}\mathbb{E}X_{k}=n\mu,\qquad\operatorname{Var}S_{n}=\sum_{k=1}^{n}\operatorname{Var}X_{k}\le nc$$

(the finite sums evaluated and compared by Finite Sums in a Commutative Ring and in an Ordered Field: Distributivity, Differences, Telescoping, Constant Terms, Comparison and the Triangle Inequality (iv) and Finite Sums in a Commutative Ring and in an Ordered Field: Distributivity, Differences, Telescoping, Constant Terms, Comparison and the Triangle Inequality (v)).

Proof of (i). Let $n\in\mathbb{N}$. As $n>0$, $|S_{n}(\omega)/n-\mu|\ge\varepsilon$ holds exactly when $|S_{n}(\omega)-n\mu|\ge n\varepsilon$, so Chebyshev's inequality (Markov's and Chebyshev's Inequalities (ii), with $a=n\varepsilon$) gives

$$P\Bigl(\Bigl|\frac{S_{n}}{n}-\mu\Bigr|\ge\varepsilon\Bigr)=P\bigl(|S_{n}-\mathbb{E}S_{n}|\ge n\varepsilon\bigr)\le\frac{\operatorname{Var}S_{n}}{n^{2}\varepsilon^{2}}\le\frac{c}{n\varepsilon^{2}}.$$

Proof of (ii). Let $a_{n}$ be the probability bounded in (i), a real number with $0\le a_{n}\le1$. Let $\eta>0$ be real. By The Integers in the Real Numbers: Greatest Elements, the Integer Part, and Archimedean Consequences (vii) there is $N\in\mathbb{N}$ with $c/(\varepsilon^{2}\eta)<n$ for every $n\ge N$, and for such $n$, by (i),

$$|a_{n}-0|=a_{n}\le\frac{c}{n\varepsilon^{2}}<\eta.$$

Hence $a_{n}\to0$ (Convergent Sequences of Real Numbers (i)).