*Independent random variables X_1, ... , X_n have normal distribution with mean 0 and variance 2. What is the distribution of random variable: X_1 + ... + X_n ?*

I know that the sum od two random variables with normal distribution has normal distribution as well, but does it apply to infinite number of variables? Besides, when we add these up, it's going to be just one random variable so shouldn't it have single-point distribution?