I am considering the following problem:
Let $F_n(\boldsymbol{\theta})$ is a sequence of random variables depending on a set of parameters $\boldsymbol{\theta}$ and the function mapping from $\boldsymbol{\theta}$ to $F_n$ is continuous. Now assume that $$F_n(\boldsymbol{\theta})\overset{P}{\to}F(\boldsymbol{\theta})$$ and $$\boldsymbol{\theta}_n\overset{P}{\to}\boldsymbol{\theta}\,$$.
Then would the following be correct? $$F_n(\boldsymbol{\theta}_n)\overset{P}{\to}F(\boldsymbol{\theta})\,$$.
If it is correct, could anyone provide a proof of this? Thanks!