I have a very beginner's question regarding the Central Limit Theorem (CLT):
I am aware that the CLT states that a mean of i.i.d. random variables is approximately normal distributed (for $n \to \infty$, where $n$ is the index of the summands) or the standardized random variable would have a standard normal distribution.
Now the Law of Large Number states roughly speaking that the mean of i.i.d random variables converges (in probability or almost surely) to their expected value.
What I don't understand is: If, as the CLT states, the mean is approximately normally distributed, how then can it also converge to the expected value at the same time?
Convergence would imply for me that with time the probability that the mean takes a value which is not the expected value is almost zero, hence the distribution would not really be a normal but almost zero everywhere except at the expected value.
Any explanation is welcome.