Convergence at infinitely many points in a compact set and convergence in the whole region
Solution 1:
It is true.
$(f_n)$ is uniformly bounded, so by Montel's theorem it is normal: Every subsequence of $(f_{n_k})$ of $(f_n)$ has a (locally uniformly) convergent subsequence $(f_{n_{k_l}})$.
On the other hand, every convergent subsequence has the same limit: Assume that $f_{n_{1, k}} \to F_1$ and $f_{n_{2, k}} \to F_2$, then $F_1(z) = F_2(z)$ for all points where the original sequence is convergent, i.e. for infinitely many points in $K$. $K$ is compact, so these points have an accumulation point in $K$, and it follows from the identity theorem that $F_1 = F_2$.
Now assume that $(f_n(a))$ is not convergent for some $a \in \Omega$. Then there exist subsequences $(f_{n_{1, k}})$ and $(f_{n_{2, k}})$ such that both $(f_{n_{1, k}(a)})$ and $(f_{n_{2, k}}(a))$ are both convergent but with a different limit. According to (1), both sequences have subsequences which are convergent in $\Omega$. According to (2), those convergent subsequences have the same limit. This is a contradiction.
The same argument can be used to show that $(f_n)$ is in fact locally uniformly convergent, i.e. uniformly convergent on all compact subsets of $\Omega$.