Convergence of a stochastic integral to a normal random variable

Question

Let $X_t$ be the Ornstein-Uhlenbeck process defined by: $$ X_t = X_0 \, e^{-t} + \int_0^t e^{-(t-s)} dW_s. $$ Is it possible to show using elementary tools, in particular without using the central limit theorem for martingales or Markov chains, that $$ I_t :=\frac{1}{\sqrt{t}} \int_0^t X_s \, dW_s $$ converges in distribution to $\mathcal N(0, 1/2)$ as $t \to \infty$?