This is the problem that I have been given:
Consider the Independent-Set problem, in which the input is an undirected graph $G = (V,E)$ and a parameter $k$, and the goal is to determine if $G$ has an independent set of size $k$. Suppose we have an oracle $O$ for solving this decision version of independent set (think of it as a library function that takes input a graph $G$ and $k$ and answers YES/NO). Prove that there exists an algorithm that can find an independent set of size $k$, if one exists, using a polynomial number of calls to the oracle $O$, and possibly a polynomial amount of computation of its own.
My first question is: Is there any way to prove that an algorithm exists without just giving a specific algorithm?
I'm sort of taking a crash course in computer science, so this is not my strongest subject. Any hints as to what direction to take this would be appreciated!