For full-dimensional cone $K$, $x\in int(K)$, can you take arb. vector $y$ and for some $t$ small enough have that $x-ty\in int(K)$

Question

I'm working on the proof of the following theorem:

$K$ full-dimensional, closed, convex cone. $x\in int(K) \iff y^Tx>0 \quad \forall y\in K^*-{0}$

And we're pretty set with the $\Leftarrow$ proof by contradiction. So we assume $x\in int(K)$ and $y^Tx=0$ for some $y\in K^*-{0}$. Then because $x\in int(K)$, we have an $\epsilon>0$ such that $B(x,\epsilon) \subset int(K)$. Then we assume that because the cone is full dimensional, we can find some $t$ small enough such that $x-ty \in B(x,\epsilon)$ and we get contradiction.

Is this allowed. Feels a bit illegal to subtract vectors from eachother in a cone.

Answer 1

I think my answer is right. Actually, you can refer to the similar corollary in this book ``convex analysis'' (Corollary 6.4.1)