Proving an algorithm exists to find independent set from a graph given an oracle 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!
If the problem has the property that the decision version can be converted to the search version, it is self reducibility. You can show that all NP-complete problems are self reducible. The idea is to build the instance with the prefix of the verifier for NP-complete problem, and solve it based on the decision version. This is kinda well-known several lines proof.

I am not aware of any other methods to show the fact you asked for.