Repeatedly sampling posteriors for Bayesian inference + convergence

I start with some prior, $$\Theta$$, with PMF $$f_{\theta}(\theta)$$. $$\theta$$ is the probability of an event, $$X$$, yielding a success. Then, I use Monte-Carlo simulation to sample a $$\theta$$ from $$\Theta$$, and then I simulate whether $$X$$ would yield a success. I then create the posterior $$f_{\Theta | X}(\theta | x_1)$$. I repeat this process: sample a $$\theta$$ from the posterior, use it simulate whether $$X$$ yields a success, and incorporate that into a new posterior, $$f_{\Theta | X} (\theta | x_1, x_2)$$. As the number of repetitions of this process goes to infinity, will this converge to the same distribution? I presume not.