Let, $ \mathbb H = (\theta_{1},\theta_{2},..,\theta_{k})$, where $'\mathbb H'$ denotes the parametric space.
Let $ X_{1},X_{2},...,X_{n} $ be $'n'$ i.i.d. observations from a common density function, $f(x|\theta)$, $\theta \in \mathbb H$.
Consider the set of priors $(p_{1},p_{2},..,p_{k})$, where $p_{i} > 0, i = 1,2,...,k$ and $\sum_{i=1}^k p_{i} = 1$.
Here, $(\theta_{1},\theta_{2},..,\theta_{k})$ are discrete, so $(p_{1},p_{2},..,p_{k})$ are their respective "probability mass functions" (p.m.f.).
Suppose $\theta_{t} \in \mathbb H$ is the true value of $\theta$.
Here, $\theta_{i}'s$ are distinguishable, such that $\forall i \neq t$, $$\int f(x|\theta) \frac {f(x|\theta_{t})} {f(x|\theta_{i})} dx > 0.$$
Then, with probability $1$, prove, $$\lim_{n\rightarrow\infty} P(\theta = \theta_{t}|x_{1},x_{2},\dots,x_{n}) = 1 $$ and $$\lim_{n\rightarrow\infty} P(\theta = \theta_{i}|x_{1},x_{2},\dots,x_{n}) = 0, \forall i \neq t.$$
How do I prove these limits ? How do I proceed ?
Thanks in advance.