Results

(i)	R2484: Moments of a Bernoulli random boolean number
(ii)	R2483: Variance of Bernoulli random boolean number

show that $\mathbb{E} X_1 = \theta$ and $\text{Var} X_1 = \theta (1 - \theta)$, whence this result is a consequence of R3843: I.I.D. real central limit theorem with the identity index sequence. $\square$