Proof using convergence of characteristic functions
By Taylor's theorem for complex functions, the characteristic function of any random variable, X, with finite mean μ, can be written as
All X1, X2, ... have the same characteristic function, so we will simply denote this φX.
Among the basic properties of characteristic functions there are
If X and y are independent.
These rules
can be used to calculate the characteristic function of 
in
terms of φX:
The limit eitμ is the characteristic function of the constant random variable μ, and hence by the Lévy continuity theorem, converges in distribution to μ:
μ is a constant, which implies that convergence in distribution to μ and convergence in probability to μ are equivalent (see Convergence of random variables.) Therefore,
This shows that the sample mean converges in probability to the derivative of the characteristic function at the origin, as long as the latter exists.