You don't have an infinite number of random variables - you have n random variables. If you know what the distribution of the sum of two normal random variables is then recognize that you can write this as

((((X1 + X2) + X3) + ... ) + Xn)

Since X1+X2 is the sum of two random normal variables you know the distribution of it. That sum itself gives you a single random variable - call it Z. Then (X1 + X2 + X3) = (X1 + X2) + X3 = Z + X3 which is the sum of two normal random variables. Use this logic to figure out the distribution of the sum of n normal random variables.