Central Limit Theorem

Let $(X_m)_{m\in\mathbb{N}}$ be a sequence of independent and identically distributed random variables on a probability space $(\Omega,\mathcal{F},P)$ such that $X_1$ and $X_1^{2}$ have finite expectation, and suppose $\sigma^{2}=\operatorname{Var}(X_1)>0$; write $\mu=\mathbb{E}[X_1]$, $\sigma$ for the positive square root of $\sigma^{2}$ (Existence and Uniqueness of the Nonnegative Square Root), and $S_n=X_1+\cdots+X_n$. Then

$$\frac{S_n-n\mu}{\sigma\sqrt{n}}\ \longrightarrow\ Z\qquad(n\to\infty)\quad\text{in distribution},$$

where $Z$ is a standard normal random variable; explicitly, in the sense of Convergence in Probability,

$$P\Bigl(\frac{S_n-n\mu}{\sigma\sqrt{n}}\le t\Bigr)\ \longrightarrow\ \Phi(t)\qquad\text{for every }t\in\mathbb{R},$$

the convergence holding at every $t$ because $\Phi$ is continuous everywhere by claim 3 of The Gaussian Integral Is Finite and Positive, and the First Two Gaussian Moments.