Sum of $n$ Poisson random variables with parameter $\frac 1 n$ I have been working through the exercises of a textbook and stumbled upon the question as follows:

A skeptic gives the following argument to show that there must be a ﬂaw in
the central limit theorem: “We know that the sum of independent Poisson random variables follows a Poisson distribution with a parameter that is the sum
of the parameters of the summands. In particular, if $n$ independent Poisson
random variables, each with parameter $n^{−1}$, are summed, the sum has a Poisson distribution with parameter $1$. The central limit theorem says that as $n$
approaches inﬁnity, the distribution of the sum tends to a normal distribution,
but the Poisson with parameter $1$ is not the normal.” What do you think of this
argument?

My first intuition is that he is wrong, but I do not quite know how to put it into words. I believe that the flaw is in assuming that the sum of $n$ terms as $n \rightarrow \infty$ will still result in a Poisson random variable with parameter $1$. But I do not quite know how to show whether this is true.
Any hints would be greatly appreciated!

The central limit theorem applies to a sequence of IID random variables with a fixed distribution.  If you have a fixed distribution for the underlying sequence of random variables then a suitably standardised version of the sample mean will converge in distribution to the standard normal so long as its variance is finite.
In the example given in your question, the distribution is dependent on $n$ so the underlying distribution for each random variable is changing as $n \rightarrow \infty$.  The example is of course a case where the average converges in distribution to something other than the normal distribution, so it effectively shows that you cannot dispense with the assumption of a fixed distribution in the CLT.
