How can I prove that the sum of $X_1, X_2, \ldots,X_n$ random variables, all of which have normal distributions $N(\mu_i, \sigma_i)$, is a random variable that is itself normally distributed with mean $$\mu =\sum_{i=1}^n \mu_i$$
and variance
$$\sigma^2 = \sum_{i=1}^n \sigma_i^2$$
Edit: I forgot to add that this was with the assumption that all $X_1, X_2,\ldots,X_n$ are independent.