Proving that recursively enumerable languages are closed against taking prefixes
Below is a hint for working with the Turing Machine (TM) formalism for RE languages. But finishing that approach from the hint depends on how you've been working with TMs.
You have a TM, say $T_L$ to accept L and you want to construct a new TM $T_L^'$ for $\text{Prefix}(L)$.  You can start $T_L^'$ on a string $x$ and then do something to finish up with the hypothetical $y$ that completes $xy\in L$. How you do that depends somewhat on the methods you have been using to work with TMs. But that's a hint so far.