property of cones and their duals

I am reading Convex Optimization by Boyd and Vandenberghe (free at http://www.stanford.edu/~boyd/cvxbook/) and I am trying to justifying their assertion (p. 53) that if $K$ is a proper cone, $K^*$ is its dual, $v \in \operatorname{int}{K}$, and $\lambda \in K^* \setminus \{0\}$, then $\lambda^T v > 0$. It is obvious that $\lambda^T v \geq 0$. I reworded their statement and got rid of one of the directions of a double implication that was obvious to me. If anyone wants, I can define all the terms I used above.

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Let $\epsilon > 0$ be small enough so that $v - \epsilon \lambda \in K$. Then $0 \leq \lambda^T(v - \epsilon\lambda) = \lambda^T v - \epsilon\|\lambda\|^2$.