Why Is the Sum of Independent Normal Random Variables Normal?

The first standard proof consists of the computation of the convolution of two normal densities to find the density of the sum of the random variables. Throughout this article we assume that our normal random variables have mean 0 since a general normal random variable can be written in the form σ Z + μ, where Z is standard normal and μ is a constant. One then finds the convolution of two normal densities to be