I am watching MIT ocw lectures by Prof. Tsitsiklis on probability (youtube link is below).
My doubt is regarding a point he makes in the lecture on the Central Limit Theorem.
He says: The Central Limit theorem is a statement about CDFs, not PMFs or PDFs.
My doubt is this: Consider a continuous random variable with a pdf that is differentiable everywhere. If we write the $$S_n = X_1 + \dots + X_n$$ and then $$Z_n = (S_n-E[S_n])/\text{standardev}(S_n),$$ Then the CDF of $Z_n$ will converge to that of the standard normal.
Now the PDF of $Z_n$ will be the derivative of the CDF of $Z_n.\,$ So the PDF of $Z_n$ should converge to that of the standard normal. Now $S_n$ is linear function of $Z_n$, so will the PDF of $S_n$ not be a normal distribution?
For reference: MIT ocw Probability About the 8 min mark.
As you say, if the variables being summed are continuous, then their sum is also continuous: you can state the CLT as a result on the convergence of the pdf.
In general, however, you need to use the cdf.
For example, if you are summing discrete variables, their sum is a discrete variable (even if the number of summands is large). In this case, the sum has a pmf, not a pdf (a convolution of pmfs, see https://www.statlect.com/glossary/convolutions). Hence, speaking of convergence of a sequence of pmfs to a pdf does not make sense.