Final answer:
To prove that there are infinitely many Maximum Likelihood Estimators (MLEs) for θ, we need to show that the likelihood function has multiple local maxima. The likelihood function is constant within the interval (θ - 1/2, θ + 1/2) for any value of θ, leading to infinitely many possible MLEs.
Step-by-step explanation:
To prove that there are infinitely many Maximum Likelihood Estimators (MLEs) for θ, we need to show that the likelihood function has multiple local maxima. The likelihood function for this problem is given by L(θ) = f(x_1) * f(x_2) * ... * f(x_n), where f(x) is the probability density function (pdf) of the random variables. In this case, the pdf is the uniform distribution, which means it is constant within the interval (θ - 1/2, θ + 1/2) and 0 outside that interval.
Since the pdf is constant within the interval, the likelihood function will also be constant within that interval for any value of θ. Therefore, any value within the interval (θ - 1/2, θ + 1/2) can be a local maximum of the likelihood function. This means there are infinitely many possible values of θ that maximize the likelihood function, and hence, there are infinitely many MLEs for θ.