Why the set of choice rule is non-empty, if a set preference relation is finite

Question

I want to prove that if there is preference relation, induced choice rule of the set is non-empty if the set is finite. I tried to solve this problem by induction. First, I considered a set with only one element. Therefore, the choice rule of the set is the only element of the set. Then I took two-element set and by definition of completeness, I could prove that the two-element set is not empty. And for the third element set, I used the definition of completeness and transitivity to prove that the set is not empty. I generalized this induction for k-1 element set. But I do not know how to connect the $kth$ element to the induction to reach the ultimate proof.