For (1) there is actually nothing to prove in my opinion. If $\displaystyle x* \in X \subset R^n$ maximizes f(x) s.t. h(x) = c then it is clear that (II) follows from (I)
For (2) imagine a function that has its absolute maximum in the negative domain of $\displaystyle R^n$ and decreases the closer it gets to zero and even decreases further for any positive number. Let this e.g. be for $\displaystyle R^1$: $\displaystyle 100 - (10 + x)^2$. If you maximize this function for x you will get $\displaystyle x* = -10$. However, if you can only choose from the positive numbers (you might want to use Kuhn Tucker to prove this formally) your $\displaystyle x* = 0$. However, this differs to the solution for (I) thus you proved that (2) is wrong by counter example.