I want to prove the following theorem: Let $D\subseteq R$ be a choice set, $F\subseteq D$ be the feasible set, $\Theta\subseteq R$ be a parameter space, and $f:D\times\Theta\to R$, an objective function. Assume that $F$ is compact, and for each $\theta\in\Theta$, the maximization problem $\max_{x\in F}f(x,\theta)$ has a unique solution. Let the rule that associates each $\theta$ with the corresponding maximizer be described by a function $x^*:\Theta\to F$, and let $v^*:\Theta\to R$ denote the value function: $v^*(\theta)\equiv f(x^*(\theta),\theta)$. Show that, if $f$ is continuous, then both $x^*$ and $v^*$ are continuous.
What I'm trying to contruct is, take any convergent sequence $(\theta)_{n\in N_+}$ in $\Theta$ and consider the sequence $(x_n^*)_{n\in N_+}$ in $X$ defined as $x_n^*\equiv x^*(\theta_n)$. But I got stuck that for any subsequence of $(x_n^*)_{n\in N_+}$, there exists a convergent sub-sequence. I also want to show that all of these convergent sub-sequences must have the same limit $x_0^*\equiv x^*(\theta_0)$. Can someone please show me to the correct way?