In the proof I have of the CLT theorem, we use the fact that, for $(X_i)_{i\geqslant 1}$ a sequence of iid random variables in $L^2(\mathbb{P})$, we have $$\lim\limits_{n\to+\infty}\Phi_{Z_n}(u)=\exp\left(-\frac{u^2\sigma^2}{2}\right)$$ where $\Phi$ is the characteristic function, and $$Z_n=\sqrt{n}\left(\overline{X_n}-\mathbb{E}\left[X_1\right]\right)=\frac{1}{\sqrt{n}}\sum_{i=1}^n Y_i$$ where $Y_i=X_i-\mathbb{E}\left[X_i\right]$.
However, it is legitimate to wonder whether there exists a random variable $Z$ such that $Z_n$ converges to $Z$ almost surely. If that is the case, because $Z_n$ converges in distribution to $\mathcal{N}(0,\sigma^2)$, then we must have $Z\sim\mathcal{N}(0,\sigma^2)$.
But I have no way to go further. I have read somewhere that we can use the sequence $$Z_n'=\frac{1}{\sqrt{n}}\sum_{i=n+1}^{2n}Y_i$$
Indeed, this converges on distribution to a random variable $G$ such that $G\sim\mathcal{N}(0,\sigma^2)$. But what is the link between $Z_n$ and $Z_n'$ ? If $Z_n$ converges almost surely to some random variable $Z$, then what is the limit (almost surely) of $Z_n'$ ?