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Define Lc = {f: f - a ∈ X* for some real number a and f(x) ≥ 0 for all x ∈ C} (X* is the dual space of X)

Using a version of the Hahn - Banach Theorem to show that

C = ∩ {x ∈ X: f(x) ≥ 0} with the index f ∈ Lc under the intersection

Could someone help me to solve this problem, i cant see how Hahn - Banach can imply the above statement ( I used the separation version to obtain g(x)<a<g(y) for some linear functional g)