you can prove that but you cannot pull the square root through the expectation.
I would just cook up an example and show that
Try a bernoulli with P(X=0)=1-p and P(X=1)=p.
Here you have pq as the pop. variance.
Let n=3 and obtain the distribution of S and see what happens.
The sample of (0,0,0) occurs with probability q times q times q.
A sample of two 0's and one 1 occurs with probability pqq, but there are 3 of those....
So you can derive the distribution of S via these four possibilities.