Finding maximum-cardinality independent set with a particular oracle We suppose we have a polynomial algorithm which receives a graph $G$ (any graph) and returns a stable set of $G, SA(G)$ with the following property:

$\alpha(G) − |SA(G)| \leq k$ , for every natural $k$ ($k$ is a constant)

$\alpha(G)$ is the stability number

I need to show that this algorithm can be used to find(in polynomial time) a stable set (of maximum cardinality) in a graph.

I discovered that if I give $k=0$ then we get to this relation. But does that mean that the algorithm finds the stable set with maximum cardinal.
If $k = 0$ then your algorithm outputs a stable set of maximum size. In the general case, try duplicating each vertex $k+1$ times, and see if it helps.