Show that a compact set in an infinite dimensional space has an empty interior.

i know that a normed space is finite dimensional iff a closed unit ball is compact. but i am not sure how to make use of this theorem.

Let K be a compact set of X, which is infinite dimensional.

Assume that int(K) is not empty.

suppose x is in K. then there is an open ball with radius r, centered x. and this ball is entirely contained in K.

how can i use the compact closed unit ball to make a contradiction? help please.