Convergence in Distribution of Sums of Random Variable

Any statement that says $\lim_{n\to\infty}(\cdots\cdots) = (\text{something depending on $n$})$ is wrong if taken literally, and usually wrong if taken any other way.

The distribution $N(n\mu,n\sigma^2)$ depends on $n$ and does not approach a limit as $n$ grows.

However, the distribution of $$ \frac{Y-n\mu}{\sigma\sqrt n} \tag 1 $$ does approach a limit as $n$ grows (unless $\sigma=+\infty,$ as happens in some cases). That limit is $N(0,1).$

This may be understood as meaning that the c.d.f. of $(1)$ converges pointwise to the c.d.f. of $N(0,1).$ If the limit were a distribution that concentrates positive probability at some points, it would be understood as meaning that the c.d.f. converges pointwise except at points where the limiting distribution assigns positive probability.